Answer:
0.813
0.500
Step-by-step explanation:
Use binomial probability.
P = nCr p^r q^(n−r)
where n is the number of trials,
r is the number of successes,
p is the probability of success,
and q is the probability of failure (1−p).
In this problem, n = 5, p = 0.5, and q = 0.5.
"At least 2 girls" means r = 2, 3, 4, or 5.
Or, we can use the complement.
P(at least 2 girls) = 1 − P(at most 1 girl)
P(at least 2 girls) = 1 − P(r=0 or r=1)
P(at least 2 girls) = 1 − ₅C₁ (0.5)¹ (0.5)⁵⁻¹ − ₅C₀ (0.5)⁰ (0.5)⁵⁻⁰
P(at least 2 girls) = 1 − 5 (0.5) (0.5)⁴ − 1 (1) (0.5)⁵
P(at least 2 girls) = 1 − 6 (0.5)⁵
P(at least 2 girls) ≈ 0.813
"At most 2 girls" means r = 0, 1, or 2.
P(at most 2 girls) = P(r=0, r=1, or r=2)
P(at most 2 girls) = ₅C₀ (0.5)⁰ (0.5)⁵⁻⁰ + ₅C₁ (0.5)¹ (0.5)⁵⁻¹ + ₅C₂ (0.5)² (0.5)⁵⁻²
P(at most 2 girls) = 1 (1) (0.5)⁵ + 5 (0.5) (0.5)⁴ + 10 (0.5)² (0.5)³
P(at most 2 girls) = 16 (0.5)⁵
P(at most 2 girls) = 0.500
