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?
5/8
To determine the probability of correctly identifying the coins, we can break down the problem into a series of steps.
Step 1: Determine the probability of getting one coin to show heads and the other to show tails in a single flip.
Since the probability of flipping heads with the fair coin is 1/2 and flipping tails with the weighted coin is 1/4, the probability of getting one coin to show heads and the other to show tails in a single flip is:
Probability = (Probability of heads with fair coin) × (Probability of tails with weighted coin)
Probability = 1/2 × 1/4
Probability = 1/8
Step 2: Determine the probability of not getting one coin to show heads and the other to show tails in a single flip.
The complement of the probability calculated in Step 1 will give us the probability of not getting one coin to show heads and the other to show tails in a single flip:
Probability = 1 - (Probability of getting one coin to show heads and the other to show tails in a single flip)
Probability = 1 - 1/8
Probability = 7/8
Step 3: Determine the probability of not correctly identifying the coins in the first flip.
The probability of not correctly identifying the coins after the first flip is equal to the probability of not getting one coin to show heads and the other to show tails in a single flip, which we calculated in Step 2.
Step 4: Determine the probability of correctly identifying the coins by repeatedly flipping until one coin shows heads and the other shows tails.
Since each flip is independent of the others, the probability of correctly identifying the coins after multiple flips is the complement of the probability of not correctly identifying the coins in the first flip raised to the power of the number of flips:
Probability = (Probability of not correctly identifying the coins in the first flip)^(Number of flips)
Probability = (7/8)^1 (since we only need one flip to identify the coins)
Therefore, the probability of correctly identifying the coins using this method is:
Probability = (7/8)^1 = 7/8 ≈ 0.875
Hence, there is approximately a 87.5% chance of correctly identifying the coins.