A survey shows that about \( 60 \% \) (i.e., 0.6 ) of all students know the HelpYourMath YouTube channel in a college. Suppose that we randomly select 10 students from the college, please solve the following problems and round your answers to 4 decimal places. a. What is the probability that exactly 3 of them know HelpYourMath YouTube Channel? b. What is the probability that at least 3 of them know HelpYourMath YouTube Channel? c. What is the probability that at most 3 of them know HelpYourMath YouTube Channel? d. What is the probability that more than 3 of them know HelpYourMath YouTube Channel? e. What is the probability that less than 3 of them know HelpYourMath YouTube Channel? ?

To solve these problems, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where: - \( n \) is the number of trials (students selected), - \( k \) is the number of successes (students who know the channel), - \( p \) is the probability of success, - \(\binom{n}{k}\) is the binomial coefficient. For this scenario: - \( n = 10 \) - \( p = 0.6 \) Let's compute the requested probabilities. ### a. Probability that exactly 3 of them know HelpYourMath YouTube Channel: \[ P(X = 3) = \binom{10}{3} (0.6)^3 (0.4)^{10-3} \] Calculating this: \[ P(X = 3) = \frac{10!}{3!(10-3)!} (0.6)^3 (0.4)^7 \] \[ = 120 * 0.216 * 0.00164 \approx 0.0392 \] ### b. Probability that at least 3 of them know HelpYourMath YouTube Channel: We calculate \( P(X \geq 3) \) as: \[ P(X \geq 3) = 1 - P(X < 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)) \] Calculating \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \): - For \( P(X = 0) \): \[ P(X = 0) = \binom{10}{0} (0.6)^0 (0.4)^{10} = (0.4)^{10} \approx 0.0001 \] - For \( P(X = 1) \): \[ P(X = 1) = \binom{10}{1} (0.6)^1 (0.4)^9 \approx 10 * 0.6 * 0.000262 = 0.0016 \] - For \( P(X = 2) \): \[ P(X = 2) = \binom{10}{2} (0.6)^2 (0.4)^8 \approx 45 * 0.36 * 0.00065536 = 0.0106 \] So, \[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.0001 + 0.0016 + 0.0106 = 0.0123 \] Then, \[ P(X \geq 3) = 1 - 0.0123 \approx 0.9877 \] ### c. Probability that at most 3 of them know HelpYourMath YouTube Channel: This is: \[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \] \[ P(X \leq 3) \approx 0.0001 + 0.0016 + 0.0106 + 0.0392 = 0.0515 \] ### d. Probability that more than 3 of them know HelpYourMath YouTube Channel: \[ P(X > 3) = 1 - P(X \leq 3) \approx 1 - 0.0515 = 0.9485 \] ### e. Probability that less than 3 of them know HelpYourMath YouTube Channel: \[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.0123 \] ### Summary of Answers - a. \( P(X = 3) \approx 0.0392 \) - b. \( P(X \geq 3) \approx 0.9877 \) - c. \( P(X \leq 3) \approx 0.0515 \) - d. \( P(X > 3) \approx 0.9485 \) - e. \( P(X < 3) \approx 0.0123 \)