Answer:
a) 0.0437
b) 0.6479
c) 0.3958
Step-by-step explanation:
This is a negative binomial distribution problem. A negative binomial distribution problem describes a sequence of trials, each of which can have two outcomes (success or failure). We continue the trials indefinitely until we get r succeses.
The probability of y trials before obtaining r successes is given as
P(y) = ʸ⁻¹Cᵣ₋₁ pʳ qʸ⁻ʳ
p = 0.15
q = 1 - p = 0.85
r = 3
y = 15, ≥15 and ≤15
a) Probability of interviewing exactly 15 applicants.
p = 0.15
q = 1 - p = 0.85
r = 3
y = 15,
P(y=15) = ¹⁵⁻¹C₃₋₁ (0.15)³ (0.85)¹⁵⁻³
P(y=15) = ¹⁴C₂ (0.15)³ (0.85)¹² = 0.0436859997 = 0.0437
b) Probability of interviewing at least 15 applicants = P(y ≥ 15)
P(y ≥ 15) = 1 - P(y < 15)
P(y < 15) = P(y=3) + P(y=4) + P(y=5) + P(y=6) + P(y=7) + P(y=8) + P(y=9) + P(y=10) + P(y=11) + P(y=12) + P(y=13) + P(y=14)
= 0.003375 + 0.00860625 + 0.014630625 + 0.0207267188 + 0.0264265664 + 0.031447614 + 0.0356406292 + 0.0389501162 + 0.0413844985 + 0.0429938957 + 0.0438537736 + 0.0440531089
P(y < 15) = 0.3520887963
P(y ≥ 15) = 1 - P(y < 15) = 1 - 0.3520887963 = 0.6479112037 = 0.6479
c) Probability of interviewing at most 15 applicants
P(y ≤ 15) = P(y=3) + P(y=4) + P(y=5) + P(y=6) + P(y=7) + P(y=8) + P(y=9) + P(y=10) + P(y=11) + P(y=12) + P(y=13) + P(y=14) + P(y=15)
(y ≤ 15) = 0.003375 + 0.00860625 + 0.014630625 + 0.0207267188 + 0.0264265664 + 0.031447614 + 0.0356406292 + 0.0389501162 + 0.0413844985 + 0.0429938957 + 0.0438537736 + 0.0440531089 + 0.0436859997 = 0.395774796 = 0.3958
Hope this Helps!!!
