A person belonging to a certain group with a probability of 0.2 brunette, 0.3 shoten, 0.4 blonde, 0.1 red.

A person belonging to a certain group with a probability of 0.2 brunette, 0.3 shoten, 0.4 blonde, 0.1 red. A group of 7 people is chosen at random. find the probability that the group has the same number of brunettes and redheads.

1. Complete group of independent events:

A1 – brunette
A2 – brown-haired,
A3 – blond
A4 – redhead.
p1 = 0.2;
p2 = 0.3;
p3 = 0.4;
p4 = 0.1.
Let be:

p = p1 + p4 = 0.3;
q = p2 + p3 = 0.7;
P = p1p4 = 0.02.
2. Suppose that among n = 7 people there are k brunettes and k redheads. Let’s use the transformed Bernoulli formula:

P (k) = C (n, 2k) * p ^ (2k) * q ^ (n – 2k) * C (2k, k) * (p1p4) ^ k.
P (0) = C (7, 0) * p ^ 0 * q ^ 7 * C (0, 0) * P ^ 0 = q ^ 7 = 0.7 ^ 7 = 0.082354;
P (1) = C (7, 2) * p ^ 2 * q ^ 5 * C (2, 1) * P ^ 1 = 21 * 0.3 ^ 2 * 0.7 ^ 5 * 2 * 0.02 ≈ 0.012706;
P (2) = C (7, 4) * p ^ 4 * q ^ 3 * C (4, 2) * P ^ 2 = 35 * 0.3 ^ 4 * 0.7 ^ 3 * 6 * 0.02 ^ 2 ≈ 0.000233;
P (3) = C (7, 6) * p ^ 6 * q ^ 1 * C (6, 3) * P ^ 3 = 7 * 0.3 ^ 6 * 0.7 ^ 1 * 20 * 0.02 ^ 3 ≈ 0.000001;
P (X) = P (0) + P (1) + P (2) + P (3) = 0.082354 + 0.012706 + 0.000233 + 0.000001 ≈ 0.0953.
Answer: 0.0953.