Bob has n coins in his pocket. One is two-headed, the rest are fair. A coin ispicked at random, lipped, and shows heads. Find the probability that the coinis two-headed

Sure, let's solve this step by step.

Step 1: Identify the given information Bob has n coins in total. One of these coins is two-headed (meaning it will always land on heads), and the rest are fair coins (meaning they have a 50% chance of landing on heads and a 50% chance of landing on tails). Bob picks one coin at random and flips it. It lands on heads. We need to find the probability that the coin he picked is the two-headed one.

Step 2: Define the events Let's define: Event A as the coin is two-headed. Event B as the coin shows heads.

Step 3: Find the probabilities of the events The probability of event A, P(A), is the probability that Bob picks the two-headed coin. Since there is only one two-headed coin out of n total coins, P(A) = 1/n.

The probability of event B, P(B), is the probability that the coin shows heads. This can happen in two ways: either Bob picks the two-headed coin (which will always show heads), or he picks a fair coin (which has a 50% chance of showing heads). So, P(B) = P(A) + (1 - P(A))/2 = 1/n + (n - 1)/2n = (1 + n - 1)/2n = n/2n = 1/2.

Step 4: Find the conditional probability We need to find the probability of event A given that event B has occurred, denoted as P(A|B). According to Bayes' theorem, P(A|B) = P(B|A) * P(A) / P(B).

P(B|A) is the probability that the coin shows heads given that the two-headed coin was picked. Since the two-headed coin will always show heads, P(B|A) = 1.

So, P(A|B) = 1 * (1/n) / (1/2) = 2/n.

So, the probability that the coin Bob picked is the two-headed one, given that it landed on heads, is 2/n.