Question

37​% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is​ (a) exactly​ three, (b) at least​ four, (c) less than eight.

Answer 1

Please note ⁿC_x = n! / [(n-x)!*x!]

Binomial Probability = ⁿC_x * (p)^x * (q)^n-x, where n = number of trials and x is the number of successes.

Also sum of probabilities from 0 till n = 1, i.eP(0) + P(1) + P(2) +.......+P(n) = 1

_____________________________________________________________

Here n = 12, p = 0.37, q = 1 – p = 0.63.

_____________________________________________________________

(a) P(Exactly 3)

P(X = 3) = ¹²C₃ * (0.37)³ * (0.63)^12-3 = 0.1742

_____________________________________________________________

(b) P(At Least 4) = P(4) + P(5) + ....... + P(12) = 1 - [P(0) + P(1) + P(2) + P(3)]

P(X = 0) = ¹²C₀ * (0.37)⁰ * (0.63)^12-0 = 0.0039

P(X = 1) = ¹²C₁ * (0.37)¹ * (0.63)^12-1 = 0.0276

P(X = 2) = ¹²C₂ * (0.37)² * (0.63)^12-2 = 0.0890

P(X = 3) = ¹²C₃ * (0.37)³ * (0.63)^12-3 = 0.1742

Therefore P(At least 4 = 1 - [0.0039 + 0.0276 + 0.089 + 0.1742] = 1 - 0.2947 = 0.7053

_____________________________________________________________

(c) P(Less than 8) = P(7) + P(6) + P(5) + ........+P(0)

We know that P(0) + P(1) + P(2) + P(3) = 0.2947

P(X = 4) = ¹²C₄ * (0.37)⁴ * (0.63)^12-4 = 0.2302

P(X = 5) = ¹²C₅ * (0.37)⁵ * (0.63)^12-5 = 0.2163

P(X = 6) = ¹²C₆ * (0.37)⁶ * (0.63)^12-6 = 0.1482

P(X = 7) = ¹²C₇ * (0.37)⁷ * (0.63)^12-7 = 0.0746

Therefore P(X < 8) = 0.2947 + 0.2302 + 0.2163 + 0.1482 + 0.0746 = 0.964

_____________________________________________________________