Question

A certain type of new business succeeds 30 % of the time. Suppose that three such businesses open.
a) What is the probability that all three businesses succeed?



b) What is the probability that no business succeeds?



c) What is the probability that exactly one business succeeds?



d) Compute the mean and standard deviation for this problem.

Answer 1

a)All 3 businesses are independent so the probability that all businesses succeed
is
P = 0.3 * 0.3 * 0.3 = 0.027

b) For each business the probability of failure is 1 - 0.3 = 0.7. Since 3
businesses are independent we get:
P = 0.7 * 0.7 * 0.7 = 0.343

c) The probability that the first business succeed but two other - fail is p1 =
0.3 * 0.7 * 0.7 = 0.147
Taking 3 different combinations (1st business succeeds, 2nd, 3d) we obtain:
P = 3 * 0.147 = 0.441

d) Let X be a random variable corresponding the number of successful
businesses. Let's construct the probability table:
X = 0, p = 0.343
X = 1, p = 0.441
X = 2, p = 1 - 0.343 - 0.441 - 0.027 = 0.189
X = 3, p = 0.027
Then the mean is:
E(X) = sum pi * Xi = 0.441 + 2 * 0.189 + 3 * 0.027 = 0.9
Standard deviation:
S(X) = sqrt(E(X-E(X))^2) = sqrt(E(X^2) - E^2(X))
E(X^2) = 0.441 + 4 * 0.189 + 9 * 0.027 = 1.44
S(X) = sqrt(1.44 - 0.9^2) = sqrt(1.44 - 0.81) = 0.79