Although cities encourage carpooling to reduce traffic congestion, most vehicles

Question

Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one person. For example, 77% of vehicles on the roads are occupied by just the driver. (Round your answers to four decimal places.)

(a) If you choose 10 vehicles at random, what is the probability that more than half (that is, 6 or more) carry just one person?

(b) If you choose 96 vehicles at random, what is the probability that more than half (that is, 49 or more) carry just one person? (Use the normal approximation.)

Explanation / Answer

(a) n = 10, p = 0.77, q = 1 - p = 0.23

P(x) = C(n, x) p^x q^(n - x)

P(6 or more) = P(6) + P(7) + P(8) + P(9) + P(10)

= C(10, 6) 0.77^6 0.23^(10 - 6) + C(10, 7) 0.77^7 0.23^(10 - 7) + C(10, 8) 0.77^8 0.23^(10 - 8)

+ C(10, 9) 0.77^9 0.23^(10 - 9) + C(10, 10) 0.77^10 0.23^(10 - 10)

= 0.1225 + 0.2343 + 0.2942 + 0.2188 + 0.0733 = 0.9431

(b) n = 96, p = 0.77, q = 1 - p = 0.23

= np = 96 * 0.77 = 73.92, = (npq) = (96 * 0.77 * 0.23) = 4.12

x = 48.5

z = (x - )/ = (48.5 - 73.92)/4.12 = -6.17

P(x > 48.5) = P(z > -6.17) = 1.00

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