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3.3 Independent and Mutually Exclusive Events

independent Events

Two events are independent if the following are true:

  • P(A|B) = P(A)
  • P(B|A) = P(B)
  • P(A AND B) = P(A)P(B)

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions.

If two events are NOT independent, then we say that they are dependent.

Sampling may be done with replacement or without replacement.

  • With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
  • Without replacement: When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.

If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise.

You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king) of that suit.

  1. Sampling with replacement:Suppose you pick three cards with replacement. The first card you pick out of the 52 cards is the

    Q of spades. You put this card back, reshuffle the cards and pick a second card from the 52-card deck. It is the ten of clubs. You put this card back, reshuffle the cards and pick a third card from the 52-card deck. This time, the card is the Q of spades again. Your picks are {Q of spades, ten of clubs, Q of spades}. You have picked the Q of spades twice. You pick each card from the 52-card deck.

  2. Sampling without replacement:Suppose you pick three cards without replacement. The first card you pick out of the 52 cards is the

    K of hearts. You put this card aside and pick the second card from the 51 cards remaining in the deck. It is the three of diamonds. You put this card aside and pick the third card from the remaining 50 cards in the deck. The third card is the J of spades. Your picks are {K of hearts, three of diamonds, J of spades}. Because you have picked the cards without replacement, you cannot pick the same card twice.

model 1

You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king) of that suit. Three cards are picked at random.

  1. Suppose you know that the picked cards are Q of spades, K of hearts and Q of spades. Can you decide if the sampling was with or without replacement?

    Show Answer

    Sampling with surrogate

  2. Suppose you know that the picked cards are Q of spades, K of hearts, and J of spades. Can you decide if the sampling was with or without replacement?

    Show Answer

    no, we can not tell if the sample distribution was with or without substitution.

case 2

You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), and K (king) of that suit. S = spades, H = Hearts, D = Diamonds, C = Clubs.

  1. Suppose you pick four cards, but do not put any cards back into the deck. Your cards are QS, 1D, 1C, QD.
  2. Suppose you pick four cards and put each card back before you pick the next card. Your cards are KH, 7D, 6D, KH.

Which of 1 or 2 did you sample with replacement and which did you sample without replacement?

Show Answer

  1. Without replacement
  2. With replacement

This video provides a brief lesson on finding the probability of independent events.

Try It

You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), and K (king) of that suit. S = spades, H = Hearts, D = Diamonds, C = Clubs. Suppose that you sample four cards.

  1. QS, 7D, 6D, KS
  2. KH, 7D, 6D, KH
  3. QS, 1D, 1C, QD

Which of the following outcomes are possible for sampling without replacement?

Show Answer

Without replacement:

  1. QS, 7D, 6D, KS: Possible
  2. KH, 7D, 6D, KH: Impossible
  3. QS, 1D, 1C, QD: Possible

Without replacement :

Which of the following outcomes are possible for sampling with replacement?

Show Answer

With replacement:

  1. QS, 7D, 6D, KS: Possible
  2. KH, 7D, 6D, KH: Possible
  3. QS, 1D, 1C, QD: Possible

With replacement :

You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J ( jack ), Q ( fagot ), and K ( king ) of that become. S = spades, H = Hearts, D = Diamonds, C = Clubs. Suppose that you sample four cards.Which of the follow outcomes are possible for sampling without substitution ? Which of the follow outcomes are possible for sampling with replacement ?

mutually single Events

A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0.

For example, suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}.

  1. A AND B = {4, 5}.

    P(A AND B)=210\displaystyle{P}{({A} \text{ AND } {B})}=\frac{{2}}{{10}}

    P

    (

    A

     AND 

    B

    )

    =

    10

    2

    and is not equal to zero. Therefore, A and B are not mutually exclusive.

  2. A AND C do not have any numbers in common so P(A AND C) = 0. Therefore, A and C are mutually exclusive.

If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms.

exemplar 3

Flip two fair coins. (This is an experiment.)

The sample space is {HH, HT, TH, TT} where T = tails and H = heads. The possible outcomes are HH, HT, TH, and TT. The outcomes HT and TH are different. The HT means that the first coin showed heads and the second coin showed tails. The TH means that the first coin showed tails and the second coin showed heads.

  • Let A = the event of getting at most one tail. (At most one tail means zero or one tail.) Then A can be written as {HH, HT,TH}. The outcome HH shows zero tails. HT and TH each show one tail.
  • Let B = the event of getting all tails. B can be written as {TT}. B is the complement of A, so B = A′.

    P(A) + P(B) = P(A) + P(A′) = 1.

  • The probabilities for A = P(event A) =

    34\frac{{3}}{{4}}

    4

    3

    .

  • Let C = the event of getting all heads. C = {HH}. Are eevnt C and event B mutually exclusive?

    Show Answer

    Since B = { TT }, P ( B AND C ) = 0.

    B and C are mutually exclusive. ( B and C have no members in common because you can not have all tails and all heads at the same clock time. )

  • Let D = event of getting more than one tail. D = {TT}. Find P(D).

    Show Answer

    14\frac{{1}}{{4}}

    4

    1

    P ( D ) =

  • Let E = event of getting a head on the first roll. (This implies you can get either a head or tail on the second roll.) Find P(E).

    Show Answer

    E = {HT,HH}.

    P(E) =

    24\frac{{2}}{{4}}

    4

    2

    E = { HT, HH } .P ( E ) =

  • Find the probability of getting at least one (one or two) tail in two flips.

    Show Answer

    Let F = event of getting at least one tail in two flips.

    F = {HT, TH, TT}.

    P(F) =

    34\frac{{3}}{{4}}

    4

    3

    Let F = event of getting at least one tail in two flips.F = { HT, TH, TT } .P ( F ) =

Try It

Draw two cards from a standard 52-card deck with replacement. Find the probability of getting at least one black card.

Show Answer

The sample space of drawing two cards with replacement from a standard 52-card deck with respect to color is {BB, BR, RB, RR}.

Event A = Getting at least one black card = {BB, BR, RB}

P(A)=34=0.75\displaystyle{P}{({A})}=\frac{{3}}{{4}}={0.75}

P

(

A

)

=

4

3

=

0.75

The sample quad of drawing two cards with surrogate from a standard 52-card deck with deference to color is { BB, BR, RB, RR } .Event A = Getting at least one total darkness card = { BB, BR, RB }

Draw two cards from a standard 52-card deck with refilling. Find the probability of getting at least one black card .

example 4

Flip two fair coins. Find the probabilities of the events.

  1. Let F = the event of getting at most one tail (zero or one tail).
  2. Let G = the event of getting two faces that are the same.
  3. Let H = the event of getting a head on the first flip followed by a head or tail on the second flip.
  4. Are F and G mutually exclusive?
  5. Let J = the event of getting all tails. Are J and H mutually exclusive?

Show Answer

The sample space is {HH, HT, TH, TT} where T = tails and H = heads.

  1. Zero (0) or one (1) tails occur when the outcomes HH, TH, HT show up. P(F) =

    34\frac{{3}}{{4}}

    4

    3

  2. Two faces are the same if HH or TT show up. P(G) =

    24\frac{{2}}{{4}}

    4

    2

  3. A head on the first flip followed by a head or tail on the second flip occurs when HH or HT show up. P(H) =

    24\frac{{2}}{{4}}

    4

    2

  4. F and G share HH so P(F AND G) is not equal to zero (0). F and G are not mutually exclusive.
  5. Getting all tails occurs when tails shows up on both coins (TT). H’s outcomes are HH and HT.

    J and H have nothing in common so P(J AND H) = 0. J and H are mutually exclusive.

The sample distribution distance is { HH, HT, TH, TT } where T = tails and H = heads .

This video provides two more examples of finding the probability of events that are mutually exclusive.

Try It

A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement).

  1. If event F = the event of getting the white ball twice, find P(F).

    Show Answer

    14\displaystyle\frac{{1}}{{4}}

    4

    1

    P ( F ) =

  2. If event G = the event of getting two balls of different colors, find P(G).

    Show Answer

    12\displaystyle\frac{{1}}{{2}}

    2

    1

    P ( G ) =

  3. If event H = the event of getting white on the first pick, find P(G).

    Show Answer

    12\displaystyle\frac{{1}}{{2}}

    2

    1

    P ( H ) =

  4. Are F and G mutually exclusive?

    Show Answer

    Yes

  5. Are G and H mutually exclusive?

    Show Answer

    No

A corner has two balls, one white and one crimson. We select one ball, put it back in the corner, and select a second ball ( sampling with substitution ) .

example 5

Roll one fair, six-sided die. The sample space is {1, 2, 3, 4, 5, 6}. Let event

A = a face is odd. Then A = {1, 3, 5}. Let event B = a face is even. Then B = {2, 4, 6}.

  • Find the complement of A, A′.

    Show Answer

    The complement of A, A′, is B because A and B together make up the sample space.

    P(A) +P(B) = P(A) + P(A′) = 1.

    Also, P(A) =

    36\displaystyle\frac{{3}}{{6}}

    6

    3

    . The complement of A, A′, is B because A and B in concert make up the sample space.P ( A ) +P ( B ) = P ( A ) + P ( A′ ) = 1.Also, P ( A ) =

  • Let event C = odd faces larger than two. Let event D = all even faces smaller than five.

    Are C and D mutually exclusive events? Why?

    Show Answer

    C = { 3, 5 } .D = { 2, 4 }.

    P ( C AND D ) = 0 because you can not have an odd and even front at the same time.

    Therefore, C and D are mutually exclusive events .

  • Let event E = all faces less than five.

    Are C and E mutually exclusive events? Why?

    Show Answer

    C = {3, 5} and E = {1, 2, 3, 4}.

    C and E = {3}

    P(C AND E) =

    16\frac{{1}}{{6}}

    6

    1

    .

    No, event C and event E are not mutually exclusive events. C = { 3, 5 } and E = { 1, 2, 3, 4 } .C and E = { 3 } P ( C AND E ) =No, event C and event E are not mutually exclusive events .

  • Find P(C|A).

    Show Answer

    This is a conditional probability.

    Recall that the event C is {3, 5} and event A is {1, 3, 5}.

    To find P(C|A), find the probability of C using the sample space A.

    You have reduced the sample space from the original sample space {1, 2, 3, 4, 5, 6} to {1, 3, 5}.

    So, P(C|A) =

    23\displaystyle\frac{{2}}{{3}}

    3

    2

    . This is a conditional probability.Recall that the event C is { 3, 5 } and event A is { 1, 3, 5 } .To find P ( C|A ), find the probability of C using the sample distribution space A.You have reduced the sample space from the original sample quad { 1, 2, 3, 4, 5, 6 } to { 1, 3, 5 } .So, P ( C|A ) =

Try It

Let event A = learning Spanish. Let event B = learning German. Then A AND B = learning Spanish and German.

Suppose P(A) = 0.4 and P(B) = 0.2. P(A AND B) = 0.08.

Are events A and B independent?

Hint:

You must show ONE of the following:

  • P(A|B) = P(A)
  • P(B|A)
  • P(A AND B) = P(A)P(B)

Show Answer

P(A|B)=

P(A AND B)P(B)=0.080.2=0.4=P(A)\frac{{{P}{({A}\text{ AND }{B})}}}{{{P}{({B})}}}=\frac{{0.08}}{{0.2}}={0.4}={P}{({A})}

P

(

B

)

P

(

A

 AND 

B

)

=

0.2

0.08

=

0.4

=

P

(

A

)

The events are independent because P(A|B) = P(A). P ( A|B ) =The events are independent because P ( A|B ) = P ( A ) .

Let event A = learning spanish. Let event B = learn German. then A AND B = learning spanish and German.Suppose P ( A ) = 0.4 and P ( B ) = 0.2. P ( A AND B ) = 0.08.Are events A and B autonomous ?

example 6

Let event G = taking a math class. Let event H = taking a science class. Then, G AND H = taking a math class and a science class. Suppose P(G) = 0.6, P(H) = 0.5, and P(G AND H) = 0.3. Are G and H independent?

Hint:If G and H are independent, then you must show ONE of the following:

  • P(G|H) = P(G)
  • P(H|G) = P(H)
  • P(G AND H) = P(G)P(H)
  • method 1 :

    Show that P(G|H) = P(G).

    P(G|H) =

    P(G and H)P(H)\frac{\text{P(G and H)}}{\text{P(H)}}

    P(H)

    P(G and H)

    =

    0.30.5\frac{0.3}{0.5}

    0.5

    0.3

    = 0.6

    P(G) = 0.6

    Since P(G|H) = P(G), G and H are independent. express that P ( G|H ) = P ( G ) .P ( G|H ) == 0.6P ( G ) = 0.6Since P ( G|H ) = P ( G ), G and H are independent .

  • method 2 :

    Show P ( G AND H ) = P ( G ) P ( H ).

    P ( G and H ) = 0.3

    P ( G ) * P ( H ) = 0.6 * 0.5 = 0.3

    Since P ( G AND H ) = P ( G ) P ( H ), G and H are autonomous .

  • method acting 3 :

    Show that P(H|G) = P(H).

    P(H|G) =

    P(H and G)P(G)\frac{\text{P(H and G)}}{P(G)}

    P

    (

    G

    )

    P(H and G)

    =

    0.30.6\frac{0.3}{0.6}

    0.6

    0.3

    = 0.5

    P(H) = 0.5

    Since P(H|G) = P(H), G and H are independent. prove that P ( H|G ) = P ( ) .P ( H|G ) == 0.5P ( ) = 0.5Since P ( H|G ) = P ( ), G and H are independent .

Since G and H are independent, knowing that a person is taking a science class does not change the chance that he or she is taking a math class. If the two events had not been independent (that is, they are dependent) then knowing that a person is taking a science class would change the chance he or she is taking math.

Try It

In a bag, there are six red marbles and four green marbles. The red marbles are marked with the numbers 1, 2, 3, 4, 5, and 6. The green marbles are marked with the numbers 1, 2, 3, and 4.

  • R = a red marble
  • G = a green marble
  • O = an odd-numbered marble
  • The sample space is S = {R1, R2, R3, R4, R5, R6, G1, G2, G3, G4}.

S has ten outcomes. What is P(G AND O)?

Show Answer

Event G and O = {G1, G3}

P(G and O) =

210\displaystyle\frac{{2}}{{10}}

10

2

= 0.2 Event G and O = { G1, G3 } P ( G and O ) == 0.2

In a bag, there are six crimson marbles and four green marbles. The red marbles are marked with the numbers 1, 2, 3, 4, 5, and 6. The green marbles are marked with the numbers 1, 2, 3, and 4.S has ten result. What is P ( G AND O ) ?

model 7

Let event C = taking an English class. Let event D = taking a speech class.

Suppose P(C) = 0.75, P(D) = 0.3, P(C|D) = 0.75 and P(C AND D) = 0.225.

Justify your answers to the following questions numerically.

  1. Are C and D independent?

    Show Answer

    Yes, because P ( C|D ) = P ( C ) .

  2. Are C and D mutually exclusive?

    Show Answer

    No, because P ( C AND D ) is not peer to zero .

  3. What is P(D|C)?

    Show Answer

    P(DmidC)=P(C AND D)P(C)=0.2250.75=0.3\displaystyle{P}{({D}{mid}{C})}=\frac{{{P}{({C} \text{ AND }{D})}}}{{{P}{({C})}}}=\frac{{0.225}}{{0.75}}={0.3}

    P

    (

    D

    mi

    d

    C

    )

    =

    P

    (

    C

    )

    P

    (

    C

     AND 

    D

    )

    =

    0.75

    0.225

    =

    0.3

Try It

A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = 0.40, P(D) = 0.30 and P(B AND D) = 0.20.

  1. Find P(B|D).
  2. Find P(D|B).
  3. Are B and D independent?
  4. Are B and D mutually exclusive?

Show Answer

  1. P(B|D) = 0.6667
  2. P(D|B) = 0.5
  3. No
  4. No

A student goes to the library. Let events B = the student checks out a book and D = the scholar checks out a DVD. Suppose that P ( B ) = 0.40, P ( D ) = 0.30 and P ( B AND D ) = 0.20 .

exemplar 8

In a box there are three red cards and five blue cards. The red cards are marked with the numbers 1, 2, and 3, and the blue cards are marked with the numbers 1, 2, 3, 4, and 5. The cards are well-shuffled. You reach into the box (you cannot see into it) and draw one card.

Let R = red card is drawn, B = blue card is drawn, E = even-numbered card is drawn.

The sample space S = R1, R2, R3, B1, B2, B3, B4, B5. S has eight outcomes.

  • Find P(R).

    Show Answer

    38\frac{{3}}{{8}}

    8

    3

    .  P ( R ) =

  • Find P(R and B).

    Show Answer

    P ( R AND B ) = 0. ( You can not draw one card that is both red and blue. )

  • Find P(E).

    Show Answer

    38\frac{{3}}{{8}}

    8

    3

    . (There are three even-numbered cards, R2, B2, and B4.) P ( E ) =. ( There are three even-numbered cards, R2, B2, and B4. )

  • Find P(E|B) .

    Show Answer

    P(E|B) =

    P(E and B)P(B)\frac{\text{P(E and B)}}{P(B)}

    P

    (

    B

    )

    P(E and B)

    =

    25\frac{{2}}{{5}}

    5

    2

    (There are five blue cards: B1, B2, B3, B4, and B5.

    Out of the blue cards, there are two even cards; B2 and B4.) P ( E|B ) = ( There are five blue cards : B1, B2, B3, B4, and B5.Out of the gloomy cards, there are two even cards ; B2 and B4. )

  • Find P(B|E).

    Show Answer

    P(E and B)P(E)\frac{\text{P(E and B)}}{P(E)}

    P

    (

    E

    )

    P(E and B)

    =

    23\frac{{2}}{{3}}

    3

    2

    .

    There are three even-numbered cards: R2, B2, and B4.

    Out of the even-numbered cards, to are blue; B2 and B4. P ( B|E ) =There are three even-numbered cards : R2, B2, and B4.Out of the even-numbered cards, to are blue ; B2 and B4 .

  • Are events R and  mutually exclusive?

    Show Answer

    The events R and B are mutually exclusive because P ( R AND B ) = 0 .

  • Let G = card with a number greater than 3. Find P(G).

    Show Answer

    28\frac{{2}}{{8}}

    8

    2

    , P(G) = P(G|H), which means that G and H are independent. G = { B4, B5 }. P ( G ) =, P ( G ) = P ( G|H ), which means that G and H are autonomous .

Try It

In a basketball arena,

  • 70% of the fans are rooting for the home team.
  • 25% of the fans are wearing blue.
  • 20% of the fans are wearing blue and are rooting for the away team.
  • Of the fans rooting for the away team, 67% are wearing blue.

Let A be the event that a fan is rooting for the away team, B be the event that a fan is wearing blue.

  1. Are the events of rooting for the away team and wearing blue independent?
  2. Are they mutually exclusive?

Show Answer

  1. P(B|A) = 0.67, P(B) = 0.25

    So P(B) does not equal P(B|A) which means that B and A are not independent (wearing blue and rooting for the away team are not independent).

  2. They are also not mutually exclusive, because P(B AND A) = 0.20 

    ≠\ne

    =

    0.

In a basketball arena, Let A be the event that a sports fan is rooting for the away team, B be the consequence that a fan is wearing blue .

exemplar 9

In a particular college class, 60% of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, 75% have long hair. Let F be the event that a student is female. Let L be the event that a student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?

  • The following probabilities are given in this example:
  • P(F) = 0.60; P(L) = 0.50
  • P(F AND L) = 0.45
  • P(L|F) = 0.75

solution 1 :

Check if P(F AND L) = P(F) * P(L).

We are given that P(F AND L) = 0.45, but P(F) * P(L) = (0.60)(0.50) = 0.30.

P(F AND L)

≠\ne

=

P(F)P(L), the events of being female and having long hair are not independent. Check if P ( F AND L ) = P ( F ) * P ( L ) .We are given that P ( F AND L ) = 0.45, but P ( F ) * P ( L ) = ( 0.60 ) ( 0.50 ) = 0.30.P ( F AND L ) P ( F ) P ( L ), the events of being female and having hanker hair are not independent .

solution 2

Check whether P(L|F) equals P(L).

We are given that P(L|F) = 0.75, but P(L) = 0.50.

Since P(L|F)

≠\ne

=

P(L), the events of being female and having long hair are not independent. Check whether P ( L|F ) equals P ( L ) .We are given that P ( L|F ) = 0.75, but P ( L ) = 0.50.Since P ( L|F ) P ( L ), the events of being female and having long hair are not independent .

interpretation of Results

The events of being female and having long hair are not independent; knowing that a student is female changes the probability that a student has long hair.

Try It

Mark is deciding which route to take to work. His choices are I = the Interstate and F = Fifth Street.

  • P(I) = 0.44 and P(F) = 0.55
  • P(I AND F) = 0 because Mark will take only one route to work.

What is the probability of P(I OR F)?

Show Answer

P ( I AND F ) = 0,

P ( I OR F ) = P ( I ) + P ( F ) – phosphorus ( I AND F ) = 0.44 + 0.56 – 0 = 1

Mark is deciding which route to take to work. His choices are I = the Interstate and F = Fifth Street.What is the probability of P ( I OR F ) ?

exemplar 10

  1. Toss one fair coin (the coin has two sides, H and T). The outcomes are ________. Count the outcomes. There are ____ outcomes.

    Show Answer

    H and T ; 2

  2. Toss one fair, six-sided die (the die has 1, 2, 3, 4, 5 or 6 dots on a side). The outcomes are ________________. Count the outcomes. There are ___ outcomes.

    Show Answer

    1, 2, 3, 4, 5, 6 ; 6

  3. Multiply the two numbers of outcomes. The answer is _______.

    Show Answer

    2 ( 6 ) = 12

  4. If you flip one fair coin and follow it with the toss of one fair, six-sided die, the answer in three is the number of outcomes (size of the sample space). What are the outcomes? (Hint: Two of the outcomes are H1 and T6.)

    Show Answer

    T1, T2, T3, T4, T5, T6, H1, H2, H3, H4, H5, H6

  5. Event A = heads (H) on the coin followed by an even number (2, 4, 6) on the die.

    A = {_________________}. Find P(A).

    Show Answer

    312\frac{{3}}{{12}}

    12

    3

    A = { H2, H4, H6 } ; P ( A ) =

  6. Event B = heads on the coin followed by a three on the die. B = {________}. Find P(B).

    Show Answer

    112\frac{{1}}{{12}}

    12

    1

    B = { H3 } ; P ( B ) =

  7. Are A and B mutually exclusive?

    (Hint: What is P(A AND B)? If P(A AND B) = 0, then A and B are mutually exclusive.)

    Show Answer

    Yes, because P ( A AND B ) = 0

  8. Are A and B independent?

    (Hint: Is P(A AND B) = P(A)P(B)? If P(A AND B) = P(A)P(B), then A and B are independent. If not, then they are dependent).

    Show Answer

    P(A AND B) = 0.

    P(A) * P(B) =

    312\frac{3}{12}

    12

    3

    *

    112\frac{1}{12}

    12

    1

    =

    3144\frac{3}{144}

    144

    3

    .

    P(A AND B)

    ≠\ne

    =

    P(A)P(B), so A and B are dependent. P ( A AND B ) = 0.P ( A ) * P ( B ) =P ( A AND B ) P ( A ) P ( B ), so A and B are dependent .

Try It

A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let T be the event of getting the white ball twice, F the event of picking the white ball first, S the event of picking the white ball in the second drawing.

  1. Compute P(T).
  2. Compute P(T|F).
  3. Are T and F independent?.
  4. Are F and S mutually exclusive?
  5. Are F and S independent?

Show Answer

  1. P(T) =

    14\frac{{1}}{{4}}

    4

    1

  2. P(T|F) =

    12\frac{{1}}{{2}}

    2

    1

  3. No
  4. No
  5. Yes

A box has two balls, one white and one loss. We select one ball, put it back in the box, and select a second ball ( sampling with successor ). Let T be the event of getting the white ball doubly, F the consequence of picking the white ball foremost, S the event of picking the white ball in the second base draw .

References

Lopez, Shane, Preety Sidhu. “U.S. Teachers Love Their Lives, but Struggle in the Workplace.” Gallup Wellbeing, 2013. http://www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx (accessed May 2, 2013).

Data from Gallup. Available online at www.gallup.com/ (accessed May 2, 2013).

Concept Review

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent.

In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.

Formula Review

If A and B are independent, P(A AND B) = P(A)P(B), P(A|B) = P(A) and P(B|A) = P(B).

If A and B are mutually exclusive, P(A OR B) = P(A) + P(B) and P(A AND B) = 0.

Licenses and Attributions

CC licensed content, Shared previously

  • OpenStax, Statistics, Individual and Mutually Exclusive Events. License: CC BY: Attribution
  • Introductory Statistics. Authored by: Barbara Illowski, Susan Dean. Provided by: Open Stax. Located at: http://cnx.org/contents/ [ electronic mail protected ]. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected]

All rights reserved content

  • Ex: Probability of Events that are Mutually Exclusive Events. Authored by: Mathispower4u. Located at: https://youtu.be/zxhDOvS2c3k. License: All Rights Reserved. License terms: Standard YouTube License
  • Probability of Independent Events. Authored by: Mathispower4u. Located at: https://youtu.be/QsnfXUqsFPU. License: All Rights Reserved. License terms: Standard YouTube LIcense

freelancer and mutually exclusive do not mean the same thing.Two events are independent if the following are true : Two events A and B areif the cognition that one occurred does not affect the find the other occurs. For exemplar, the outcomes of two roles of a fair die are freelancer events. The consequence of the first roll does not change the probability for the consequence of the second roll. To show two events are independent, you must showof the above conditions.If two events are NOT freelancer, then we say that they areSampling may be doneorIf it is not known whether a and B are mugwump or dependent, You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J ( jack ), Q ( fagot ), K ( king ) of that suit.You have a honest, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are 13 cards in each courtship consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J ( jack ), Q ( queen ), K ( king ) of that suit. Three cards are picked at random.You have a fairly, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each befit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J ( jack ), Q ( fagot ), and K ( king ) of that suit. S = spades, H = Hearts, D = Diamonds, C = Clubs.Which of 1 or 2 did you sample with substitute and which did you sample without refilling ? This television provides a brief lesson on finding the probability of independent events.A and B areevents if they can not occur at the same time. This means that A and B do not share any outcomes and P ( A AND B ) = 0.For case, suppose the sample distribution distance S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } .Let A = { 1, 2, 3, 4, 5 }, B = { 4, 5, 6, 7, 8 }, and C = { 7, 9 } .If it is not known whether vitamin a and B are mutually exclusive, assume they are not until you can show otherwise. The trace examples illustrate these definitions and terms.Flip two fair coins. ( This is an experiment. ) The sample space is { HH, HT, TH, TT } where T = tails and H = heads. The possible outcomes are HH, HT, TH, and TT. The outcomes HT and TH are unlike. The HT means that the inaugural coin showed heads and the second coin showed tails. The TH means that the beginning coin showed tails and the irregular mint showed heads.Flip two fair coins. Find the probabilities of the events.This video provides two more examples of finding the probability of events that are mutually exclusive.Roll one fair, six-sided die. The sample distribution space is { 1, 2, 3, 4, 5, 6 }. Let eventA = a face is curious. then A = { 1, 3, 5 }. Let event B = a face is even. then B = { 2, 4, 6 } .Let event G = taking a mathematics classify. Let event H = taking a skill class. then, G AND H = taking a mathematics course and a science class. Suppose P ( G ) = 0.6, P ( H ) = 0.5, and P ( G AND H ) = 0.3. Are G and H freelancer ? Since G and H are freelancer, knowing that a person is taking a skill class does not change the opportunity that he or she is taking a mathematics class. If the two events had not been independent ( that is, they are dependent ) then knowing that a person is taking a skill class would change the gamble he or she is taking math.Let event C = taking an english class. Let event D = taking a manner of speaking class.Suppose P ( C ) = 0.75, P ( D ) = 0.3, P ( C|D ) = 0.75 and P ( C AND D ) = 0.225.Justify your answers to the follow questions numerically.In a box there are three crimson cards and five blue cards. The red cards are marked with the numbers 1, 2, and 3, and the blue cards are marked with the numbers 1, 2, 3, 4, and 5. The cards are well-shuffled. You reach into the box ( you can not see into it ) and draw one card.Let R = red menu is drawn, B = blue circuit board is drawn, E = even-numbered menu is drawn.The sample space S = R1, R2, R3, B1, B2, B3, B4, B5. S has eight outcomes.In a especial college class, 60 % of the students are female. Fifty percentage of all students in the class have long hair. forty-five percentage of the students are female and have long hair’s-breadth. Of the female students, 75 % have long haircloth. Let F be the event that a scholar is female. Let L be the consequence that a scholar has long hair. One student is picked randomly. Are the events of being female and having long hair independent ? Lopez, Shane, Preety Sidhu. “ U.S. Teachers Love Their Lives, but Struggle in the Workplace. ” Gallup Wellbeing, 2013. hypertext transfer protocol : //www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx ( access May 2, 2013 ) .Data from Gallup. available on-line at www.gallup.com/ ( accessed May 2, 2013 ) .Two events A and B are independent if the cognition that one occurred does not affect the prospect the early occur. If two events are not independent, then we say that they are dependent.In sampling with successor, each member of a population is replaced after it is picked, so that member has the hypothesis of being chosen more than once, and the events are considered to be autonomous. In sampling without substitute, each penis of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.If A and B are independent, P ( A AND B ) = P ( A ) P ( B ), P ( A|B ) = P ( A ) and P ( B|A ) = P ( B ) .If A and B are mutually exclusive, P ( A OR B ) = P ( A ) + P ( B ) and P ( A AND B ) = 0 .

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