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Probability of victory in this coin toss game?

\$ \begingroup \$ I ‘m new to learning probability and am a bite stummped on this elementary scenario : I wondered if I could walk you through all my unlike intelligent, so you can point out what I ‘ve misunderstood about the different techniques of probability that thread into working out something like this ?
Two players are taking it in turns to flip a mint, scoring 1 point if it lands on mind, 0 if chase – thus an adequate opportunity each musician will score a point, per go. beginning one to score 5 points wins the round .
As it stands, Player 1 has 4 points, and Player 2 has 3 – it ‘s P1 ‘s move : what is the probability that P1 will go on to win the tournament, and besides for P2 ?

here are my thoughts and where I ‘m stick :
1 ) clearly, P1 has 50 % prospect of scoring the 1 point he needs to win the turn on his future become, but 50 % surely is n’t the answer – what if the round off played to 100, P1 had 99 points and P2 good 1 – calm it ‘s 50/50 P1 will get the winning steer on the following sound, but overall it ‘s far more likely P1 will go onto gain without P2 always catching up, than fair 50 % ?
2 ) I tried next to think of it like this – the alone direction for P2 to win is to score two consecutive heads, and P1 of class to score a tail in between, leading ( with P1 going first ) to the vector : THTH. now, the probability of that accurate combination is ( 0.5 ) ^4 = P ( Player 2 Wins ). Hence, P ( Player 1 Wins ) = 1 – ( 0.5^4 ) .
3 ) But then, I again realised that this is n’t compensate – preferably than THTH being the alone vector of scores leading to P2 succeed, you could of course have any of an infinite number, each more unlikely than the last, leading to P2 winning – TTTTTHTH for case .
so, some kind of integration would be required to sum the probability that P2 would win, over all infinite cases of diminishing probability – subtract that from 1, and you have P ( P1 ). Or do it the early way around to find the probability of each, of course .
Is my assessment that 3 is the necessitate scheme, correct – i.e. some kind of integration between 0 tosses to inifinity tosses is required ?

Allow me to just return to 2 ) and modify the scenario a bit – say rather of each flipping a mint, with a stern meaning no one scores a point that go, say a reviewer flips the coin, and heads is a steer for P1 tails a point for P2 .
This wholly changes it now, and should make it simpler – now every start, one of the players is gauranteed to score a point. now we have a finite count of paths to victory ( and probability vectors ) for each player .
1 ) But still I ‘m a bit confused : with the scores at P1 : 4 & P2 : 3, the only way for P2 to win is by scoring two square tails : ( TT ). The probabiltiy the referee tosses two tails is ( 0.5 ) ^2 = 0.25. Hence P ( P2 ) = 0.25, and P ( P1 ) = 1-0.25 = 0.75 .
But, this bare scenario has a sample distance of these outcomes : ( H ), ( TH ), ( TT ). I.e. those are the merely sequences in which the referee could toss the coin and end the game .
2 ) But of those outcomes, 2 of them mean victory for P1 – sol looking at it like this, P ( P1 ) = 2/3 ? ?
here are my thoughts on this, and what I believe is the final chastise answer : I forgot that those 3 potential scenarios in the sample space are not evenly probably. ( H ) at 0.5 is doubly adenine likely as ( TH ) or ( TT ). If then you account for the burden of their likelihood, and alternatively have a newfangled pseudo-Sample quad of ( H ), ( H ), ( TH ), ( TT ) and consider each of those to be evenly likely, then indeed 3/4 scenarios mean victory for P1, which agrees with 1-P ( P2 ) = 1- ( 0.5 ) ^2 = 0.75.

Is this concluding rendition discipline ?
Thanks very much for reading through my long question ! very appreciate it, as I wanted to lay out all my thoughts on the different probabilty ideas so you can point out what I ‘m understanding correctly, and where I ‘m going incorrectly .
many thanks indeed ! very appreciate it .

reference : https://leowiki.com
Category : Economy