This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
BIG, let's knock this out. The random variable X follows a binomial distribution, X B(n, p), where n=8 (number of trials) and p=1/4 (probability of success). The probability mass function is given by: P(X=k) = nk p^k (1-p)^n-k Here, n=8, p=1/4, and 1-p = 1 - 1/4 = 3/4. a) Find P(X=2) Step 1: Identify the values for n, k, and p. For P(X=2), we have n=8, k=2, p=1/4, and 1-p=3/4. Step 2: Substitute the values into the binomial probability formula. P(X=2) = 82 ((1)/(4))^2 ((3)/(4))^8-2 P(X=2) = 82 ((1)/(4))^2 ((3)/(4))^6 Step 3: Calculate the binomial coefficient 82. 82 = (8!)/(2!(8-2)!) = (8!)/(2!6!) = (8 × 7)/(2 × 1) = 28 Step 4: Calculate the powers and multiply. P(X=2) = 28 × ((1)/(16)) × ((3^6)/(4^6)) P(X=2) = 28 × (1)/(16) × (729)/(4096) P(X=2) = (28 × 729)/(16 × 4096) = (7 × 729)/(4 × 4096) = (5103)/(16384) P(X=2) ≈ 0.31146 The probability is (5103)/(16384). b) Find P(X 1) Step 1: Use the complement rule. It is easier to calculate P(X 1) as 1 - P(X < 1). Since X is a discrete random variable, P(X < 1) is equivalent to P(X=0). P(X 1) = 1 - P(X=0) Step 2: Calculate P(X=0). For P(X=0), we have n=8, k=0, p=1/4, and 1-p=3/4. P(X=0) = 80 ((1)/(4))^0 ((3)/(4))^8-0 P(X=0) = 1 × 1 × ((3)/(4))^8 P(X=0) = (3^8)/(4^8) = (6561)/(65536) Step 3: Subtract P(X=0) from 1. P(X 1) = 1 - (6561)/(65536) P(X 1) = (65536 - 6561)/(65536) = (58975)/(65536) P(X 1) ≈ 0.90003 The probability is (58975)/(65536). What's next? Send 'em!