Weekly supply to car dealer A follows a Poisson distribution with mean $\frac{4}{3}$ cars, and weekly supply to car dealer B independently follows a Poisson distribution with mean $\frac{2}{3}$ cars. What is the probability that, in a given week, fewer than three cars in total are supplied to A and B combined?

0.406

0.677

0.857

0.920

Explanation

The combined supply follows

\text{Po}(4/3+2/3)=\text{Po}(2)

P(Y<3)=e^{-2}[1+2+2]=5e^{-2}\approx0.677

. Omitting the

P(2)

term gives

e^{-2}(3)\approx0.406

. Including

P(3)

as well (using

Y\leq3

) gives

\approx0.857

. Averaging the two means instead of adding them, using

\text{Po}(1)

, gives

e^{-1}(2.5)\approx0.920

Derived from ZIMSEC Maths Paper 1, June 2007, Q6 (Statistics)

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