A fair coin is flipped four times. What is the probability that tails occurs exactly 3 times if it is known that tails occurs at least twice?

Where did you get this question from?

The last part of the sentence " if it is known that tails occurs at least twice?" contradicts the first part.

ignoring that last part we would have:
C(4,3)(1/2)^3 (1/2)
= 4 (1/16) = 1/4

To find the probability that tails occurs exactly 3 times, given that tails occurs at least twice in four coin flips, we need to use conditional probability.

Let's break down the problem step by step:

Step 1: Determine the total number of outcomes when flipping a fair coin four times.
In each flip, there are two possible outcomes, either heads or tails. Since each flip is independent, the total number of outcomes is 2 * 2 * 2 * 2 = 16.

Step 2: Determine the number of outcomes in which tails occurs at least twice.
To find this, we need to calculate the number of outcomes with exactly 2 tails and also the number of outcomes with exactly 3 tails.

Number of outcomes with exactly 2 tails:
In four coin flips, if exactly 2 are tails, we need to choose 2 out of 4 flips to be tails. This can be done using the combination formula: C(4, 2) = 4! / (2! * (4 - 2)!) = 6.

Number of outcomes with exactly 3 tails:
In four coin flips, if exactly 3 are tails, we need to choose 3 out of 4 flips to be tails. This can be calculated using the combination formula: C(4, 3) = 4! / (3! * (4 - 3)!) = 4.

So, the total number of outcomes with tails occurring at least twice is 6 + 4 = 10.

Step 3: Determine the number of outcomes with tails occurring exactly 3 times.
Since we know that tails occurs at least twice, we need to subtract the outcomes with exactly 2 tails from the total outcomes with tails occurring at least twice. Thus, the number of outcomes with tails occurring exactly 3 times is 10 - 6 = 4.

Step 4: Calculate the probability.
To calculate the probability, we divide the number of successful outcomes (tails occurring exactly 3 times) by the total number of outcomes (16).
Therefore, the probability is 4/16 = 1/4 = 0.25.

Hence, the probability that tails occurs exactly 3 times, given that tails occurs at least twice, is 0.25 or 1/4.