For the experiment described below, let x determine a random variable, and use your knowledge of probability to prepare a probability distribution. Three cards are drawn (with replacement) and the number of red cards is noted.

To create a probability distribution for the random variable \( X \), which represents the number of red cards drawn when three cards are drawn with replacement, we first need to define the parameters of the experiment. Assuming a standard deck of cards, there are 52 cards in total, with 26 red cards (hearts and diamonds) and 26 black cards (clubs and spades). Since the cards are drawn with replacement, the probability of drawing a red card on any single draw is: \[ P(\text{Red}) = \frac{26}{52} = \frac{1}{2} \] Similarly, the probability of drawing a black card is: \[ P(\text{Black}) = \frac{26}{52} = \frac{1}{2} \] ### Random Variable \( X \) The random variable \( X \) can take on the values 0, 1, 2, or 3, representing the number of red cards drawn in three draws. ### Probability Distribution Since the draws are independent and follow a binomial distribution, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where: - \( n = 3 \) (the number of trials), - \( k \) is the number of successes (red cards drawn), - \( p = \frac{1}{2} \) (the probability of drawing a red card), - \( \binom{n}{k} \) is the binomial coefficient. Now we can calculate the probabilities for each possible value of \( X \): 1. **For \( X = 0 \)** (no red cards): \[ P(X = 0) = \binom{3}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^{3} = 1 \cdot 1 \cdot \frac{1}{8} = \frac{1}{8} \] 2. **For \( X = 1 \)** (one red card): \[ P(X = 1) = \binom{3}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{2} = 3 \cdot \frac{1}{2} \cdot \frac{1}{4} = \frac{3}{8} \] 3. **For \( X = 2 \)** (two red cards): \[ P(X = 2) = \binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{1} = 3 \cdot \frac{1}{4} \cdot \frac{1}{2} = \frac{3}{8} \] 4. **For \( X = 3 \)** (three red cards): \[ P(X = 3) = \binom{3}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{0} = 1 \cdot \frac{1}{8} \cdot 1 = \frac{1}{8} \] ### Summary of the Probability Distribution The probability distribution of \( X \) is as follows: \[ \begin{array}{|c|c|} \hline X & P(X) \\ \hline 0 & \frac{1}{8} \\ 1 & \frac{3}{8} \\ 2 & \frac{3}{8} \\ 3 & \frac{1}{8} \\ \hline \end{array} \] This distribution shows the probabilities of drawing 0, 1, 2, or 3 red cards when drawing three cards with replacement from a standard deck.