59.9k views
3 votes
Sally has two coins. The first coin is a fair coin and the second coin is biased. The biased coin comes up heads with probability .75 and tails with probability .25. She selects a coin at random and flips the coin ten times. The results of the coin flips are mutually independent. The result of the 10 flips is: T,T,H,T,H,T,T,T,H,T. What is the probability that she selected the biased coin?

1 Answer

0 votes

Final answer:

To find the probability that Sally selected the biased coin, we can use Bayes' theorem. The probability is approximately 0.7745, or 77.45%.

Step-by-step explanation:

To find the probability that Sally selected the biased coin, we can use Bayes' theorem. Let's define the following events:

  • A: Sally selected the biased coin
  • B: The result of the 10 flips is T,T, H,T,H, T,T,T, H,T

We want to find P(A|B), which is the probability that Sally selected the biased coin given the result of the 10 flips. Bayes' theorem states that:

P(A|B) = P(B|A) * P(A) / P(B)

First, let's calculate P(B|A). Since the biased coin comes up heads with a probability 0.75 and tails with a probability 0.25, the probability of getting T,T,H,T,H,T,T,T,H,T when using the biased coin is: 0.25 * 0.25 * 0.75 * 0.25 * 0.75 * 0.25 * 0.25 * 0.75 * 0.25 * 0.75 = 0.0007575

Next, let's calculate P(A) and P(B). P(A) represents the probability that Sally selected the biased coin, which is 1/2 since there are 2 coins in total. P(B) represents the probability of getting the result T,T,H,T,H,T,T,T,H,T with any coin, which is: 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5 = 0.0009765625

Now we can substitute these values into Bayes' theorem:

P(A|B) = 0.0007575 * (1/2) / 0.0009765625 ≈ 0.7745

Therefore, the probability that Sally selected the biased coin given the result of the 10 flips is approximately 0.7745, or 77.45%.

User Sergio Cabral
by
8.4k points