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What is the probability of getting 3 heads on tossing a coin three times? – GeeksforGeeks

What is the probability of getting 3 heads on tossing a coin three times ? A branch of mathematics that deals with the happening of a random event is termed probability. It is used in Maths to predict how likely events are to happen. The probability of any consequence can only be between 0 and 1 and it can besides be written in the form of a percentage .

The probability of event A is generally written as P(A).

here P represents the hypothesis and A represents the event. It states how likely an consequence is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for surely i.e. certainty

If we are not sure about the consequence of an event, we take help oneself of the probabilities of certain outcomes—how probably they occur. For a proper understanding of probability we take an exemplar as tossing a coin : There will be two possible outcomes—heads or tails. The probability of getting heads is half. You might already know that the probability is half/half or 50 % as the event is an equally probable consequence and is complementary so the possibility of getting heads or tails is 50 %. Formula of Probability

probability of an event, P ( A ) = favorable outcomes / full number of outcomes

Some Terms of Probability Theory

  • Experiment: An operation or trial done to produce an outcome is called an experiment.
  • Sample Space: An experiment together constitutes a sample space for all the possible outcomes. For example, the sample space of tossing a coin is head and tail.
  • Favorable Outcome: An event that has produced the required result is called a favorable outcome.  For example, If we roll two dice at the same time then the possible or favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).
  • Trial: A trial means doing a random experiment.
  • Random Experiment: A random experiment is an experiment that has a well-defined set of outcomes. For example, when we toss a coin, we would get ahead or tail but we are not sure about the outcome that which one will appear.
  • Event: An event is the outcome of a random experiment.
  • Equally Likely Events: Equally likely events are rare events that have the same chances or probability of occurring. Here The outcome of one event is independent of the other. For instance, when we toss a coin, there are equal chances of getting a head or a tail.
  • Exhaustive Events: An exhaustive event is when the set of all outcomes of an experiment is equal to the sample space.
  • Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either cold or hot. We cannot experience the same weather again and again.
  • Complementary Events: The Possibility of only two outcomes which is an event will occur or not. Like a person will eat or not eat the food, buying a bike or not buying a bike, etc. are examples of complementary events.

Some Probability Formulas

Addition rule: Union of two events, say A and B, then

P ( A or B ) = P ( A ) + P ( B ) – P ( A∩B ) P ( A ∪ B ) = P ( A ) + P ( B ) – P ( A∩B )

Complementary rule: If there are two possible events of an experiment so the probability of one event will be the Complement of another event. For example – if A and B are two possible events, then

P ( B ) = 1 – P ( A ) or P ( A ’ ) = 1 – P ( A ). P ( A ) + P ( A′ ) = 1 .

Conditional rule: When the probability of an event is given and the second is required for which foremost is given, then P ( B, given A ) = P ( A and B ), P ( A, given B ). It can be frailty versa

P ( B∣A ) = P ( A∩B ) /P ( A )

Multiplication rule: Intersection of two other events i.e. events A and B need to occur simultaneously. then

P ( A and B ) = P ( A ) ⋅P ( B ). P ( A∩B ) = P ( A ) ⋅P ( B∣A )

What is the probability of getting 3 heads on tossing a coin three times?

Solution:

For this first base we need to consider all the possible outcomes if we flip a coin 3 times. Outcomes will be { THH, HTH, HHT, TTH, THT, HTT, TTT, HHH } full number of result = 8 So the favorable result of having three heads = HHH = 1 therefore the probability of getting at least three heads = Probability of an event = Favorable outcomes / Total number of outcomes P ( A ) = golden outcomes / total number of outcomes = 1/8

Similar Questions Question 1: What is the probability of flipping one coin?

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Solution:

To calculate the probability of the event, It contains merely one chemical element and sample space contains two elements, so the sample space will be { H, T } therefore, sum number of consequence is 2. What is the probability of a coin landing on tails or drumhead ? The probability of landing on question is given as : P ( A ) = favorable outcomes / sum number of outcomes = 1/2 like for the probability of landing on tail, P ( A ) = friendly outcomes / sum number of outcomes = 1/2

Question 2: What are the chances of flipping 20 tails in a row? Solution:

probability of an event = ( number of friendly event ) / ( total number of event ). P ( B ) = ( occurrence of Event B ) / ( total issue of consequence ). probability of getting one tail = 1/2. here, tossing a mint is an independent event, its not dependent on how many times it has been tossed. probability of getting 3 tails in a row = probability of getting buttocks first clock × probability of getting tail second fourth dimension × probability of getting stern third base time probability of getting 3 tails in a row = ( 1/2 ) × ( 1/2 ) × ( 1/2 ) therefore, the probability of getting 20 tails in a row = ( 1/2 ) 20

Question 3: What is the probability of flipping a coin 20 times and getting 20 tails? Solution:

20 times coin tosses. This means, total observations = 220 ( According to binomial concept ) Required result → 20 tails { H, H, H, H, H, H, H, H, H, H, H, H, H, H, H, H, H, H, H, H, } This can occur only once ! frankincense, required result =1 now put the probability formula Probability ( 20 Tails ) = ( 1⁄2 ) 20 = 1⁄1048576 Hence, the probability that it will always land on the TAIL side will be, ( 1⁄2 ) 20 = 1⁄1048576

My Personal Notes

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