Solving the Probability Puzzle of Three Coins in a Box

There are three 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 three coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin? 

Probability puzzles are always fun to solve. They challenge our logical thinking and mathematical skills. In this blog post, we will discuss a probability problem that involves three coins in a box. We will analyze the problem step by step to find out the probability of selecting a specific coin from the box.

To solve this problem, we will use Bayes' theorem, which states that the probability of an event A given that event B has occurred is equal to the probability of event B given that event A has occurred, multiplied by the prior probability of event A, divided by the prior probability of event B.

Let's define the events:

H: The outcome of the flip is heads

T: The outcome of the flip is tails

TH: The coin selected is two-headed

F: The coin selected is fair

B: The coin selected is biased

Now, we need to find the probability that the coin selected was two-headed given that the outcome of the flip was heads. Mathematically, we can represent this as P(TH|H).

Using Bayes' theorem, we can write:

P(TH|H) = P(H|TH) * P(TH) / [P(H|TH) * P(TH) + P(H|F) * P(F) + P(H|B) * P(B)]

We know that P(TH) = 1/3 (as there are three coins in the box and one of them is two-headed). Also, P(H|TH) = 1 (as the two-headed coin will always show heads). Therefore, we can simplify the above equation as:

P(TH|H) = 1 * 1/3 / [1 * 1/3 + 1/2 * 1/3 + 0.75 * 1/3]

We know that P(H|F) = 1/2 (as the fair coin has an equal probability of showing heads or tails), and P(B) = 1/3 (as there are three coins in the box and one of them is biased). Also, P(H|B) = 0.75 (as the biased coin will show heads 75 percent of the time).

Plugging in the values, we get:

P(TH|H) = 1/3 / [1/3 + 1/6 + 0.25 * 1/3] = 0.4

Therefore, the probability that the coin selected was two-headed given that the outcome of the flip was heads is 0.4.

In this blog post, we have solved the probability problem of three coins in a box. We have used Bayes' theorem to calculate the probability of selecting a specific coin from the box given the outcome of the flip. The answer is 0.4, which means that there is a 40 percent chance that the selected coin was two-headed. This problem demonstrates how probability theory can help us make informed decisions in various real-life situations.