Question

Suppose you have an experiment where you flip a coin three times. You then count the number of heads. Find the probability distribution for x, the number of heads in three flips of a coin.

Answer 1

The probability distribution for the number of heads in three flips of a coin can be calculated using the binomial distribution. Each flip of the coin has a 0.5 probability of resulting in heads. Using the binomial probability formula, we can determine the probabilities for each possible outcome.

Step-by-step explanation:

When flipping a coin three times, we can determine the probability distribution for the number of heads using the concept of binomial distribution. In this case, the possible values for x (the number of heads) are 0, 1, 2, or 3. Each flip of the coin has a 0.5 probability of resulting in heads and a 0.5 probability of resulting in tails. We can use the binomial probability formula to calculate the probability of each outcome:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

Where n is the number of trials (in this case, 3), k is the number of successful outcomes (the number of heads), C(n, k) is the combination formula (n choose k), and p is the probability of a successful outcome (0.5 for heads).

Using this formula, we can calculate the probabilities for x = 0, 1, 2, and 3. The probability distribution is as follows:

1. P(X = 0) = C(3, 0) * 0.5^0 * 0.5^(3-0) = 1 * 1 * 0.125 = 0.125
2. P(X = 1) = C(3, 1) * 0.5^1 * 0.5^(3-1) = 3 * 0.5 * 0.25 = 0.375
3. P(X = 2) = C(3, 2) * 0.5^2 * 0.5^(3-2) = 3 * 0.25 * 0.5 = 0.375
4. P(X = 3) = C(3, 3) * 0.5^3 * 0.5^(3-3) = 1 * 0.125 * 1 = 0.125

Therefore, the probability distribution for x (the number of heads) in three flips of a coin is:

• x = 0 with P(X = 0) = 0.125
• x = 1 with P(X = 1) = 0.375
• x = 2 with P(X = 2) = 0.375
• x = 3 with P(X = 3) = 0.125