Final answer:
a) The probability that exactly 4 out of 5 smartphone owners have an Android is 0.178. b) The probability that 3 or fewer smartphone owners have an iPhone is approximately 0.998.
Step-by-step explanation:
a) To find the probability that exactly 4 out of 5 smartphone owners have an Android, we can use the binomial distribution formula:
P(X=k) = C(n, k) * p^k * q^(n-k)
Where n is the number of trials, k is the number of successes, p is the probability of success, and q is the probability of failure. In this case, n = 5, k = 4, p = 0.48 (the complement of 0.52), and q = 0.52. Plugging these values into the formula, we get:
P(X=4) = C(5, 4) * (0.48)^4 * (0.52)^(5-4) = 5 * 0.48^4 * 0.52 = 0.178
Therefore, the probability that exactly 4 out of 5 smartphone owners have an Android is 0.178.
b) To find the probability that 3 or fewer smartphone owners have an iPhone, we need to find the sum of the probabilities of having 0, 1, 2, or 3 iPhones. We can use the binomial distribution formula similarly to part a:
P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)
Calculating each term separately:
P(X=0) = C(5, 0) * (0.52)^0 * (0.48)^5 = 1 * 1 * 0.48^5 = 0.105
P(X=1) = C(5, 1) * (0.52)^1 * (0.48)^4 = 5 * 0.52 * 0.48^4 = 0.311
P(X=2) = C(5, 2) * (0.52)^2 * (0.48)^3 = 10 * 0.52^2 * 0.48^3 = 0.376
P(X=3) = C(5, 3) * (0.52)^3 * (0.48)^2 = 10 * 0.52^3 * 0.48^2 = 0.206
Adding up these probabilities:
P(X≤3) = 0.105 + 0.311 + 0.376 + 0.206 = 0.998
Therefore, the probability that 3 or fewer smartphone owners have an iPhone is approximately 0.998.