Probability Puzzle of Urns and Coins

Have you ever faced a probability puzzle that seemed to be unsolvable? The question of selecting a ball from two urns based on the outcome of a fair coin can be quite daunting. However, with the right approach, this puzzle can be solved with ease. In this blog post, we will explore how to solve this problem and understand the probability of selecting a ball from one urn.

First, let's understand the problem. We have two urns, urn 1 and urn 2, with different numbers of white and black balls. We flip a fair coin and select a ball from either urn based on the outcome of the coin toss. We are given that a white ball is selected and we need to find the probability that the coin landed tails.

To solve this problem, we need to use Bayes' theorem. Bayes' theorem states that the probability of an event happening, given that another event has already occurred, can be calculated by multiplying the probability of the second event happening given the first event, by the probability of the first event happening, and dividing the result by the probability of the second event happening. In this case, we need to calculate the probability of the coin landing tails given that a white ball was selected.

Let's use the following notations to represent our events:

A: Coin landed tails

B: White ball selected

U1: Ball selected from urn 1

U2: Ball selected from urn 2

Using Bayes' theorem, we can write:

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

We know that the probability of the coin landing heads or tails is the same, so P(A) = P(H) = 0.5. We also know that if the coin lands heads, we will select a ball from urn 1, and if it lands tails, we will select a ball from urn 2. Therefore, P(U1|H) = 1 and P(U2|T) = 1. Now we need to calculate P(B), the probability of selecting a white ball.

P(B) = P(B|H) * P(H) + P(B|T) * P(T)

To calculate P(B|H), the probability of selecting a white ball given that the coin landed heads, we need to add the probabilities of selecting a white ball from urn 1 and urn 2, multiplied by the probability of the coin landing heads. This gives us:

P(B|H) = P(U1) * 0.5 + P(U2) * 0.5

= (5/12) * 0.5 + (3/15) * 0.5

= 0.3125

To calculate P(B|T), the probability of selecting a white ball given that the coin landed tails, we follow a similar process. This gives us:

P(B|T) = P(U2) * 0.5 + P(U1) * 0.5

= (3/15) * 0.5 + (5/12) * 0.5

= 0.3125

Now we can calculate P(B) as:

P(B) = P(B|H) * P(H) + P(B|T) * P(T)

= 0.3125 * 0.5 + 0.3125 * 0.5

= 0.3125

Finally, we can calculate P(A|B), the probability of the coin landing tails given that a white ball was selected.