Problem 43 - A First Course in Probability

Question

There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

Step-by-Step Solution

Answer

The probability that the two-headed coin was selected, given that a head was flipped, is approximately 0.444 or 44.4%.

Step 1: Understand Bayes' Theorem

Bayes' Theorem states that for events A and B: P(A|B) = \(\frac{P(B|A) * P(A)}{P(B)}\) In our case, event A is selecting the two-headed coin, and event B is flipping a head. We want to find P(A|B), which is the probability of selecting the two-headed coin given that a head was flipped.

Step 2: Calculate the probabilities

Let's calculate the probabilities we need for Bayes' Theorem: 1. P(A): The probability of selecting the two-headed coin is 1/3, since there are three coins and each has an equal chance of being selected. 2. P(B|A): If we have selected the two-headed coin (event A), the probability of flipping a head (event B) is 100%, or 1. 3. P(B): The probability of flipping a head, considering all the coins: P(B) = P(B|A) * P(A) + P(B|A') * P(A') + P(B|A'') * P(A'') where A', A'' are the other two coins (fair coin and biased coin) We already have the probability of flipping a head given that the two-headed coin was selected, P(B|A) = 1. Now let's find the probabilities for the other coins: - Fair coin (A'): P(B|A') = 0.5 (since it's a fair coin) and P(A') = 1/3 (equal chance of being selected). - Biased coin (A''): P(B|A'') = 0.75 (heads 75% of the time) and P(A'') = 1/3 (equal chance of being selected). Now, we can calculate the overall probability of flipping a head: P(B) = P(B|A) * P(A) + P(B|A') * P(A') + P(B|A'') * P(A'') = 1 * (1/3) + 0.5 * (1/3) + 0.75 * (1/3)

Step 3: Apply Bayes' Theorem

Now that we have all the probabilities, we can use Bayes' Theorem to find the probability of selecting the two-headed coin given that a head was flipped: P(A|B) = \(\frac{P(B|A) * P(A)}{P(B)}\) P(A|B) = \(\frac{1 * (1/3)}{1*(1/3) + 0.5*(1/3) + 0.75*(1/3)}\)

Step 4: Simplify the expression

Simplifying the expression from Step 3, we will find the probability that the two-headed coin was selected: P(A|B) = \(\frac{1*(1/3)}{1*(1/3) + 0.5*(1/3) + 0.75*(1/3)}\) = \(\frac{1}{1 + 0.5 + 0.75}\) = \(\frac{1}{2.25}\) P(A|B) ≈ 0.444 So, the probability that the two-headed coin was selected, given that a head was flipped, is approximately 0.444 or 44.4%.