Q. 9.1 - A First Course in Probability

Question

Customers arrive at a bank at a Poisson rate λ. Suppose that two customers arrived during the first hour. What is the probability that 

(a) both arrived during the first 20 minutes? 

(b) at least one arrived during the first 20 minutes?

Step-by-Step Solution

Answer

The probability that (a) both arrived during the first 20 minutes is $\frac{1}{9}$ and

(b) at least one arrived during the first 20 minutes is$\frac{5}{9}$.

Part (a) Step 1: Given Information

We have given  Poisson rate is $\lambda$.

We need to find the probability that both arrived during the first 20 minutes.

Part (a) Step 2: Simplify

The conditional probability of $N(1/3)=2$, $N\left(1\right)=2$. It is equivalent to  $N(1/3)=2, N(2/3)=0$, so by independence, we have

$$\begin{gathered} P\left(N\right(1/3)=2\overline{)N\left(1\right)=2)=\frac{P\left(N\right(1/3)=2,N(1)=2)}{P\left(N\right(1)=2)}} \\ =\frac{P\left(N\right(1/3)=2,N(2/3)=0)}{P\left(N\right(1)=2)} \\ =\frac{{\left({\displaystyle \frac{\lambda }{3}}\right)}^2\frac{1}{2!}{e}^{-{\displaystyle \frac{\lambda }{3}}}.{\left({\displaystyle \frac{2\lambda }{3}}\right)}^0\frac{1}{0!}{e}^{-{\displaystyle \frac{2\lambda }{3}}}}{{\lambda }^2{\displaystyle \frac{1}{2!}}{e}^{-\lambda }}=\frac{1}{9} \end{gathered}$$

Part (b) Step 1: Given Information

We have given  Poisson rate is $\lambda$.

We need to find the probability that at least one arrived during the first 20 minutes.

Part (b) Step 2: Simplify

Similarly from the above part, we have that

$$\begin{gathered} P\left(N\right(1/3)=0\overline{)N\left(1\right)=2)=\frac{P\left(N\right(1/3)=0,N(1)=2)}{P\left(N\right(1)=2)}} \\ =\frac{P\left(N\right(1/3)=0,N(2/3)=2)}{P\left(N\right(1)=2)} \\ =\frac{{\left({\displaystyle \frac{\lambda }{3}}\right)}^0{\displaystyle \frac{1}{0!}}{e}^{-{\displaystyle \frac{\lambda }{3}}}.{\left({\displaystyle \frac{2\lambda }{3}}\right)}^2{\displaystyle \frac{1}{2!}}{e}^{-{\displaystyle \frac{2\lambda }{3}}}}{{\lambda }^2{\displaystyle \frac{1}{2!}}{e}^{-\lambda }}=\frac{4}{9} \end{gathered}$$

So, the conditional probability that some person arrived during the first$20$ minutes in which $2$ people has arrived within the first hour is $\frac{5}{9}$.