Q.4.63 - A First Course in Probability

Question

People enter a gambling casino at a rate of $1$ every $2$ minutes.

$\left(a\right)$ What is the probability that no one enters between $12:00$ and $12:05$?

$\left(b\right)$ What is the probability that at least $4$ people enter the casino during that time? 

Step-by-Step Solution

Answer

$\left(a\right)$The probability that no one enters between

$12:00$ and $12:05$ found to be $0.082$.

$\left(b\right)$The probability that at least $4$people enter the casino during that time is $0.242$,

Step 1: Given Information (Part a)

People enter a gambling casino at a rate of $1$ every $2$ minutes. 

Step 2: Calculation (Part a)

Define random variable $X$ as the number of people that enter into the casino between $12:00$ and $12:05$. Since we know that they enter with rate $1$ in every two minutes, they also enter with rate $2.5$ between given time. So we have that $X~Pois(2.5)$, the required probability is

$P(X=0)=e^{-2.5}=0.082$.

Step 3: Final answer (Part a)

The probability that no one enters between $12:00$ and $12:05$ is $0.082$.

Step 4:Given information (Part b)

People enter a gambling casino at a rate of $1$ every $2$ minutes. 

Step 5:Calculation (Part b)

The required probability is simply

$P(X\ge 4)=1-P(X=0)-P(X=1)-P(X=2)-P(X=3)$

$=1-e^{-2.5}-2.5e^{-2.5}-\frac{2.5^2}{2}{e}^{-2.5}-\frac{2.5^3}{3!}{e}^{-2.5}$

$=0.242$.

Step 6: Final answer (Part b)

The probability that at least $4$ people enter the casino during that time is $0.242$.