91Ó°ÊÓ

We need to use the binomial probability formula to calculate the probability of each given value of \(x\). The binomial formula is given by \(P(x) = \binom{n}{x}p^x q^{n-x}\), where \(n\) is the total number of trials (in this case, 5 treated seeds), \(x\) is the number of successes, and \(p\) and \(q\) are the probabilities of success and failure, respectively. a. Find the probability that \(x=4\). We want to find the probability that 4 out of the 5 treated seeds result in emerging plants. Using the binomial formula, we get: \(P(x=4) = \binom{5}{4} \cdot \left( \frac{4}{5} \right)^4 \cdot \left( \frac{1}{5} \right)^1\)

In order to find the probability, we need to evaluate the binomial coefficient \(\binom{5}{4}\). This is equal to \(\frac{5!}{(4!(5-4)!)} = 5\). So: \(P(x=4) = 5 \cdot \left( \frac{4}{5} \right)^4 \cdot \left( \frac{1}{5} \right)^1\)

Now, we just need to plug in the values and calculate the probability: \(P(x=4) = 5 \cdot (0.4096) \cdot (0.2) = 0.4096\) That means there is a 40.96% chance that 4 treated seeds will result in emerging plants. b. Find \(P(x \leq 3)\). To find the probability that 3 or fewer treated seeds result in emerging plants, we need to calculate the binomial probability for each value of \(x\) from 0 to 3, and then add them: \(P(x \leq 3) = P(x=0) + P(x=1) + P(x=2) + P(x=3)\) We can use the binomial formula for each \(x\) value, as shown in step 1: \(P(x \leq 3) = \sum_{x=0}^{3} \binom{5}{x} \cdot \left( \frac{4}{5} \right)^x \cdot \left( \frac{1}{5} \right)^{(5-x)}\) After calculating each probability and summing them, we get: \(P(x \leq 3) = 0.032 + 0.256 + 0.4096 + 0.2048 = 0.9024\) There is a 90.24% chance that 3 or fewer treated seeds will result in emerging plants. c. Find \(P(2 \leq x \leq 3)\). To find the probability that 2 to 3 treated seeds result in emerging plants, we need to calculate the binomial probability for \(x=2\) and \(x=3\), and then add them: \(P(2 \leq x \leq 3) = P(x=2) + P(x=3)\) We can use the binomial formula for each \(x\) value, as shown in step 1: \(P(2 \leq x \leq 3) = \binom{5}{2} \cdot \left( \frac{4}{5} \right)^2 \cdot \left( \frac{1}{5} \right)^3 + \binom{5}{3} \cdot \left( \frac{4}{5} \right)^3 \cdot \left( \frac{1}{5} \right)^2\) After calculating each probability and summing them, we get: \(P(2 \leq x \leq 3) = 0.4096 + 0.2048 = 0.6144\) There is a 61.44% chance that 2 to 3 treated seeds will result in emerging plants.