You and I play the following game: I toss a coin repeatedly. The coin is unfair and P(H) = p.The game ends the first time that two consecutive heads (HH) or two consecutive tails (TT) areobserved. I win if (HH) is observed and you win if (TT) is observed. Given that I won the game,find the probability that the first coin toss resulted in head
Question
You and I play the following game: I toss a coin repeatedly. The coin is unfair and P(H) = p.The game ends the first time that two consecutive heads (HH) or two consecutive tails (TT) areobserved. I win if (HH) is observed and you win if (TT) is observed. Given that I won the game,find the probability that the first coin toss resulted in head
Solution
To solve this problem, we need to consider the possible sequences of coin tosses that could lead to a win for "I" (the person tossing the coin).
-
The first possibility is that the first two tosses are heads (HH). The probability of this happening is p^2.
-
The second possibility is that the first toss is a tail, the second is a head, and the third is a head (THH). The probability of this happening is (1-p)*p^2.
Given that "I" won the game, these are the only two possibilities for the sequence of coin tosses. Therefore, the probability that the first coin toss resulted in a head is the sum of the probabilities of these two sequences.
So, the probability that the first coin toss resulted in a head given that "I" won the game is p^2 / (p^2 + (1-p)*p^2).
Simplifying this expression gives us the final answer: 1 / (1 + (1-p)).
Similar Questions
if two players A and B are alternatively throwing a coin and a die together. A player who first throws head and 6 wins the game. If A starts the game, then the probability that B wins the game is
A coin is tossed three times. An outcome is represented by a string of the sort HTT (meaning a head on the first toss, followed by two tails). The 8 outcomes are listed in the table below. Note that each outcome has the same probability.For each of the three events in the table, check the outcome(s) that are contained in the event. Then, in the last column, enter the probability of the event.Outcomes ProbabilityHHT HTH HTT HHH THH TTH THT TTTEvent A: Alternating head and tail (with either coming first) Event B: More heads than tails Event C: A tail on the second toss
Two unbiased coins are tossed. What is the probability of getting at most one head?
Two unfair coins are tossed with the probability of a tail equal to 0.25. Calculate the probability ofgetting:(a) No heads.(b) Exactly one head.(c) At least one head.
Two coins are tossed, the probability of getting at least one head is
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.