Final answer:
To find the probability of a specific number of adults favoring the use of unmanned drones, we can use the binomial probability formula. (a) The probability of exactly three adults favoring the use of drones is approximately 0.2909. (b) The probability of at least four adults favoring the use of drones is approximately 0.3752. (c) The probability of less than eight adults favoring the use of drones is approximately 0.9053.
Step-by-step explanation:
To find the probability that a specific number of adults out of a randomly selected group of twelve favor the use of unmanned drones by police agencies, we can use the binomial probability formula. The formula is:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
where P(X=k) is the probability of exactly k adults favoring the use of unmanned drones, n is the total number of adults in the group, p is the probability of an adult favoring the use of unmanned drones (32% or 0.32 in this case), and C(n, k) is the number of combinations of n things taken k at a time.
(a) To find the probability of exactly three adults favoring the use of unmanned drones, we substitute k=3, n=12, and p=0.32 into the formula:
P(X=3) = C(12, 3) * 0.32^3 * (1-0.32)^(12-3)
= 220 * 0.32^3 * 0.68^9 ≈ 0.2909 (rounded to four decimal places)
(b) To find the probability of at least four adults favoring the use of unmanned drones, we need to calculate the probability of four, five, six, ..., twelve adults favoring the use of unmanned drones, and add them together:
P(X≥4) = P(X=4) + P(X=5) + P(X=6) + ... + P(X=12)
= C(12, 4) * 0.32^4 * 0.68^8 + C(12, 5) * 0.32^5 * 0.68^7 + C(12, 6) * 0.32^6 * 0.68^6 + ... + C(12, 12) * 0.32^12 * 0.68^0 ≈ 0.3752 (rounded to four decimal places)
(c) To find the probability of less than eight adults favoring the use of unmanned drones, we need to calculate the probability of zero, one, two, ..., seven adults favoring the use of unmanned drones, and add them together:
P(X<8) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=7)
= C(12, 0) * 0.32^0 * 0.68^12 + C(12, 1) * 0.32^1 * 0.68^11 + C(12, 2) * 0.32^2 * 0.68^10 + ... + C(12, 7) * 0.32^7 * 0.68^5 ≈ 0.9053 (rounded to four decimal places)