posted by Sam on .
Anna and bob play a game in which Anna begins by rolling a fair dice, after which bob tosses a fair coin. They take turns until one of them wins. Anna wins when she rolls a 6. Bob wins when the coin lands on heads. What is the probability that Anna will win the game?
Well there are a number of ways for Anna to win like 1st it is 1/6 the second way can be 5/6*1/2*5/6*1/2*1/6 etc. Where do i go from here to find an exact number for the probability?
What happens if Anna rolls the 6 and Bob tosses a head? It it considered a tie, or does one or the other automatically win? This will change the probability of Anna winning the game.
So, if Ana rolls a 6, does the game end instantly, or does Bob get to flip a coin one last time? If he does, and gets a head, does the game continue or do they declare it a tie?
Does the question really ask for the probability that Ana wins the game, or does it ask for the probability that she wins the game within a certain number of rounds?
On Ana's first roll, she has a 1/6 chance of getting a 6.
In order to get to a second roll, she needs to roll something other than a 6 on her first roll, and Bob needs to get a tails. The probability of that happening is 5/6 * 1/2 = 5/12. Then she has a 1/6 chance of getting a 6 on her 2nd roll, so the probability that Ana wins on her second roll is 5/12*1/6 = 5/72.
probability that she wins on her third roll is 5/6*1/2*5/6*1/2*1/6=25/864
So far, the probability that she wins in three rolls or less is 1/6 + 5/72 + 25/864
This looks like a geometric series, with r=5/12.
Sum of an infinite geometric series is a1/(1-r) where a1= the first term = 1/6
a1/(1-r)=(1/6) / (7/12) = 2/7