You have two coins that look identical, but one of them is fair and the other is weighted. The weighted coin has a 3/4 chance of flipping heads and a 1/4 chance of flipping tails.
Unfortunately, you've forgotten which coin is which! You decide to keep flipping them together, one in each hand, until you get a flip where one coin shows heads and the other shows tails. Then you'll assume that the coin showing heads is the weighted coin. If you do this, what's the probability you'll correctly identify the coins?
To find the probability of correctly identifying the coins, let's break down the problem into smaller steps.
Step 1: Identify the possible outcomes of a coin flip.
- When flipping a fair coin, there are two possible outcomes: heads (H) and tails (T).
- When flipping a weighted coin, there are also two possible outcomes, but with different probabilities: heads (H) with a 3/4 chance and tails (T) with a 1/4 chance.
Step 2: Determine the possible outcomes of flipping the coins together.
Since you are flipping the coins together until one coin shows heads and the other shows tails, there are four possible outcomes:
1. HH: Both coins show heads.
2. TT: Both coins show tails.
3. HT: One coin shows heads, and the other shows tails.
4. TH: One coin shows tails, and the other shows heads.
Step 3: Calculate the probability of each outcome.
For a fair coin, the probability of each outcome is as follows:
- Probability of HH: 1/2 * 1/2 = 1/4
- Probability of TT: 1/2 * 1/2 = 1/4
- Probability of HT: 1/2 * 1/2 = 1/4
- Probability of TH: 1/2 * 1/2 = 1/4
For the weighted coin, the probabilities are different:
- Probability of HH: 3/4 * 3/4 = 9/16
- Probability of TT: 1/4 * 1/4 = 1/16
- Probability of HT: 3/4 * 1/4 = 3/16
- Probability of TH: 1/4 * 3/4 = 3/16
Step 4: Find the probability of correctly identifying the coins.
To correctly identify the coins, we need either the HT or TH outcome. So, we'll calculate the sum of those probabilities:
- Probability of HT: 1/4
- Probability of TH: 3/16
Adding these probabilities together, we get 1/4 + 3/16 = 4/16 + 3/16 = 7/16.
Therefore, the probability of correctly identifying the coins by flipping them together is 7/16 or approximately 0.4375, meaning you have a 43.75% chance of being correct.