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Chapter 4: Q.4.63 (page 167)

People enter a gambling casino at a rate of1every 2minutes.

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

(b)What is the probability that at least4people enter the casino during that time?

Short Answer

Expert verified

(a)The probability that no one enters between

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

(b)The probability that at least 4people enter the casino during that time is0.242,

Step by step solution

01

Given Information (Part a)

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

02

Calculation (Part a)

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

P(X=0)=e-2.5=0.082.

03

Final answer (Part a)

The probability that no one enters between 12:00and12:05is0.082.

04

Step 4:Given information (Part b)  

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

05

Step 5:Calculation (Part b)

The required probability is simply

P(X≥4)=1-P(X=0)-P(X=1)-P(X=2)-P(X=3)

=1-e-2.5-2.5e-2.5-2.522e-2.5-2.533!e-2.5

=0.242.

06

Final answer (Part b)

The probability that at least 4people enter the casino during that time is0.242.

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Most popular questions from this chapter

Let N be a nonnegative integer-valued random variable. For nonnegative values aj, j Ú 1, show that

∑j=1∞a1+…+ajP{N=j}=∑i=1∞aiP[N≥i}

Then show that

E[N]=∑i=1∞P[N≥i}

and

E[N(N+1)]=2∑i=1∞iP{N≥i}

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