Complete question is;
I. If 40% of all commuters ride to work in carpools, if eight workers are selected, find the probability that exactly five will ride in carpools.
II. Traffic lights with a turning arrow are red 70% of the time. If you approach 6 lights in a row, find the probability of stopping: a. Exactly four times. b. At least three times.
Answer:
I)P(X = 5) = 0.1239
IIa)P(X = 4) = 0.3241
IIb)P(X ≥ 3) = 0.9294
Step-by-step explanation:
I) This is a binomial probability problem.
Formula is;
P(X = x) = nCx(p^(x) × (1 - p)^(n - x)
40% of all commuters ride to work in carpools. Thus, p = 0.4
eight workers are selected and we are to find the probability that exactly five will ride in carpools.
Thus, n = 8 and x = 5
P(X = 5) = 8C5 × 0.4^(5) × (1 - 0.4)^(8 - 5)
P(X = 5) = 56 × 0.01024 × 0.6³
P(X = 5) = 0.1239
II) Traffic lights with a turning arrow are red 70% of the time and red means stop.
Thus; p = 0.7
Since 6 lights in a row;
n = 6
a) Probability of stopping exactly 4 times;
P(X = 4) = 6C4 × 0.7⁴ × (1 - 0.7)^(6 - 4)
P(X = 4) = 15 × 0.2401 × 0.09
P(X = 4) = 0.3241
b) probability of at least 3 times is;
P(X ≥ 3) = P(3) + P(4) + P(5) + P(6)
P(3) = 6C3 × 0.7³ × (1 - 0.7)^(6 - 3)
P(3) = 0.1852
P(4) = 0.3241
P(5) = 6C5 × 0.7^(5) × (1 - 0.7)^(6 - 5)
P(5) = 0.3025
P(6) = 6C6 × 0.7^(6) × (1 - 0.7)^(6 - 6)
P(6) = 0.1176
P(X ≥ 3) = 0.1852 + 0.3241 + 0.3025 + 0.1176
P(X ≥ 3) = 0.9294
