Question

On the Ipswich Motorway near the Warrego highway turnoff, it is known that 34% of vehicles exceed the speed limit. If 12 vehicles pass a hidden camera, what is the probability that more than three will be booked for speeding? (3 decimal places)

Answer 1

This is a Binomial Distribution with Probability of success (p) = 0.34 and Probability of failure (q) = 1 - p = 1 - 0.34 = 0.66

P(X = x) = ⁿC_x.p^x.q^{(12 - x)}

P(X = x) = ¹²C_x.(0.34)^x.(0.66)^{(12 - x)}

Therefore,

P(X > 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]

P(X = 0) = ¹²C₀.(0.34)⁰.(0.66)^{(12 - 0)} = 1 x 1 x (0.66)¹² = 1 x 0.0068 = 0.0068

P(X = 1) = ¹²C₁.(0.34)¹.(0.66)^{(12 - 1)} = 12 x 0.34 x (0.66)¹¹ = 4.08 x 0.01035 = 0.0422

P(X = 2) = ¹²C₂.(0.34)².(0.66)^{(12 - 2)} = 66 x 0.1156 x (0.66)¹⁰ = 7.63 x 0.01568 = 0.1196

P(X = 3) = ¹²C₃.(0.34)³.(0.66)^{(12 - 3)} = 220 x 0.0393 x (0.66)⁹ = 8.646 x 0.0238 = 0.2058

Hence,

P(X > 3) = 1 - (0.0068 + 0.0422 + 0.1196 + 0.2058) = 1 - 0.3744 = 0.6256

P(X > 3) = 0.626

NOTE: ⁿc_x = [n! / {x!(n - x)!}]