how do i work this out?

Question

. On a certain road section passed by 180 cars hourly 7% of drivers are speeding. What is the probability that a speed camera place on this section of the road for two hours captures a) 24 cases of speeding, b) no more than 24 cases of speeding?

Answer 1

There are a lot of moving parts here. We have to assume that the red light camera captures all speeders. Given that we are talking about a rate of arrival and not exactly 2 * 180 cars in the two hours, we might as well just look at the rate of speeders per hour equals 180 * 7 = 12.6/hour for two hours equals 25.2

Then, we can use the Poisson distribution.

To get the probability of exactly 24, you use the probability mass function rather than the probability distribution function, or exact rather than cumulative. Using λ for the arrival rate and x for the number of arrivals, t

The formula is e^-λ λ^x/x!, which would be e^-25.2 25.2²⁴/24! in this case. In Excel, we use

exp(-25.2)* 25.2^24/fact(24) = 0.0788290630491417. However, in Excel or a statistical calculator, the Poisson distribution is on it, so you just use this rather than the formula. In Excel, you use =POISSON(24,25.2,0) - 0r poisson.dist with the same values if you prefer.

The computer is especially handy when you want no more than 24, or the cumulative. You just change to using cumulative - in Excel by changing the last value to 1.

=POISSON(24,25.2,1) gives 0.457561357508727

You would expect slightly less than 1/2, as 24 is slightly less than the mean, 25.2.