On a certain road 20% of the vehicles are trucks, 16% are buses and the remainder are cars.
(i) A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses. [3]
(ii) A random sample of 125 vehicles is now taken. Using a suitable approximation, find the
probability that more than 73 are cars.
2 answers
A truck driver estimates that a country road rises 40 cm for every 6 m along the road. What is the angle of inclination of the road?
solution, P(X < 3) = P(0) + P(1) + P(2) M1
Binomial term with 11Cr p
= (0.84)11 + (0.16)(0.84)10 × 11C1 +
(0.16)2
(0.84)9
× 11C2
Correct expression for P(0, 1, 2) or P(0, 1, 2, 3)
= 0.1469 + 0.30782 + 0.2931
= 0.748
ii) μ = 125 × 0.64 = 80
σ^2= 125 × 0.64 × 0.36 = 28.8
P(X > 73) = 1 –phi(73.5-80/sqrt(28.8)
standardising = phi(1.211)
observe region, correct region (> 0.5 if mean > 73.5, if mean < 73.5)
= .887 ans
Binomial term with 11Cr p
= (0.84)11 + (0.16)(0.84)10 × 11C1 +
(0.16)2
(0.84)9
× 11C2
Correct expression for P(0, 1, 2) or P(0, 1, 2, 3)
= 0.1469 + 0.30782 + 0.2931
= 0.748
ii) μ = 125 × 0.64 = 80
σ^2= 125 × 0.64 × 0.36 = 28.8
P(X > 73) = 1 –phi(73.5-80/sqrt(28.8)
standardising = phi(1.211)
observe region, correct region (> 0.5 if mean > 73.5, if mean < 73.5)
= .887 ans
1 answer