This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
This is a binomial probability problem. Let n be the number of people selected, so n=4. Let p be the probability that a person speaks Twi, so p = 15\% = 0.15. Let q be the probability that a person does not speak Twi, so q = 1 - p = 1 - 0.15 = 0.85. The probability of exactly k successes in n trials is given by the binomial probability formula: P(X=k) = nk p^k q^n-k where nk = (n!)/(k!(n-k)!). a) more than half speak Twi "More than half" of 4 people means X > 2. So we need to find P(X=3) + P(X=4). Step 1: Calculate P(X=3). P(X=3) = 43 (0.15)^3 (0.85)^4-3 P(X=3) = (4!)/(3!1!) (0.15)^3 (0.85)^1 P(X=3) = 4 × (0.003375) × (0.85) P(X=3) = 0.011475 Step 2: Calculate P(X=4). P(X=4) = 44 (0.15)^4 (0.85)^4-4 P(X=4) = (4!)/(4!0!) (0.15)^4 (0.85)^0 P(X=4) = 1 × (0.00050625) × 1 P(X=4) = 0.00050625 Step 3: Sum the probabilities. P(X > 2) = P(X=3) + P(X=4) P(X > 2) = 0.011475 + 0.00050625 P(X > 2) = 0.01198125 Rounding to four decimal places: P(X > 2) ≈ 0.0120 b) at most 2 speak Twi "At most 2" means X 2. So we need to find P(X=0) + P(X=1) + P(X=2). Alternatively, we can use the complement rule: P(X 2) = 1 - P(X > 2). Step 1: Use the complement rule with the result from part (a). P(X 2) = 1 - P(X > 2) P(X 2) = 1 - 0.01198125 P(X 2) = 0.98801875 Rounding to four decimal places: P(X 2) ≈ 0.9880 3 done, 2 left today. You're making progress.