Question

Let \(X\) equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of \(X\) and compute the probability that \(X\) is equal to an odd number.

Short answer

The pmf of \(X\) is calculated for \(k = 0, 1, 2, 3, 4\), and then the probability of \(X\) being an odd number is found by summing the probabilities for \(k = 1, 3\).

Step-by-step solution

Define the distribution

A coin flip is a Bernoulli trial where the probability of success (getting heads, in this case) and failure (getting tails) are equal to \(0.5\). If the coin is flipped independently four times, the number of successful outcomes (\(X\)) follows a binomial distribution \(X \sim B(n, p)\), where \(n\) is the number of trials (4 coin flips) and \(p\) is the probability of success (0.5).

Identify the pmf

The pmf of a binomial distribution is given by \(P(X=k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k}\), where \(C(n, k)\) is the binomial coefficient, which gives the number of ways of choosing \(k\) successes out of \(n\) trials.

Calculate pmf for each \(k\)

We need to compute the pmf for \(k = 0, 1, 2, 3, 4\), which represents the number of heads when flipping the coin four times.

Get the probability of \(X\) being an odd number

To find the probability that \(X\) is equal to an odd number, we just need to sum the probabilities calculated in step 3 for \(k = 1, 3\).