Chapter 13, Problem 51AP

a

Summary Introduction

Interpretation:

Probability of cab to run without breakdown

Concept Introduction:

Probability is the likelihood of an event to occur.. It ranges between 0 to 1. O implies no chance of occurance while 1 implies 100% chance of occurance.

Answer to Problem 51AP

Probability of cab to run without breakdown is 0.1353.

Explanation of Solution

Given information:

Number of cabs = 10

Number of cabs breakdown = 2

The probability that any particular cab will run for a full year without suffering a breakdown is as follows:

  $\lambda \left(failurerateofanitemchosenatrandom\right)=2$

Time period (t) = 1 year

Therefore,

  $\begin{array}{l}WhenP\left(T>1\right)=e^{-\mu }\\ =e^{-2}\\ =0.1353\end{array}$

b

Summary Introduction

Interpretation:

Probability that entire fleet will run without breakdown a month.

Concept Introduction:

Probability is the likelihood of an event to occur.. It ranges between 0 to 1. O implies no chance of occurance while 1 implies 100% chance of occurance.

Answer to Problem 51AP

Probability that entire fleet will run without breakdown a month is 0.1889.

Explanation of Solution

Given information:

Number of cabs = 10

Number of cabs breakdown = 2

The probability that the entire fleet will run for one month without a breakdown is as follows:

  $\lambda \left(failurerateofanitemchosenatrandom\right)=2$

Time period (t) = 1/12 year

  $\begin{array}{l}P\left(T>1/12\right)=e^{-\mu }\\ =e^{-2/12}\end{array}$

Calculate the probability that there is no breakdown in one month as shown below:

  $\begin{array}{l}P\left(Nobreakdownfor1month\right)={\left(e^{-2/12}\right)}^{10}\\ =e^{\raisebox{1ex}{$-10$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\\ =0.1889\end{array}$

c

Summary Introduction

Interpretation:

Number of breakdown in 3 months period time.

Concept Introduction:

Probability is the likelihood of an event to occur.. It ranges between 0 to 1. O implies no chance of occurance while 1 implies 100% chance of occurance.

Answer to Problem 51AP

Number of breakdown in 3 months period time is 5.

Explanation of Solution

Given information:

Number of cabs = 10

Number of cabs breakdown = 2

The average no. of breakdowns that are expected in the typical three month period is as follows:

  $\lambda \left(failurerateofanitemchosenatrandom\right)=\frac{2}{12}\times 3$

  $=0.5perthreemonths$

The number of breakdowns expected for the entire fleet is calculated below:

Expected no. of breakdown = Total no. of cars $\times$ Rate of failure

  $\begin{array}{l}=10\times 0.5\\ =5\end{array}$

d

Summary Introduction

Interpretation:

Probability of more than 5 breakdown from Thanksgiving day and New year’s day

Concept Introduction:

Probability is the likelihood of an event to occur.. It ranges between 0 to 1. O implies no chance of occurance while 1 implies 100% chance of occurance.

Answer to Problem 51AP

Probability of more than 5 breakdown from Thanksgiving day and New year’s day is 0.

Explanation of Solution

Given information:

Number of cabs = 10

Number of cabs breakdown = 2

Formula to calculate the required probability is shown below:

  $P\left[N\left(t\right)=k\right]=e^{-2t}{\left(2t\right)}^{k/k}$

The probability that there are more than five breakdowns between Nov 28 and Jan 1 (35days) of the same year is depending on the Poisson process with $\lambda =2$ as follows:

  $P\left[N\left(\frac{35}{365}\right)>5\right]=1-{\displaystyle \sum _{i=0}^{5}P\left[N\left(35/365\right)=i\right]}$

  $=1-e^{-70/365}-\frac{70}{365}{e}^{-70/365}-\frac{{\left(70/365\right)}^2}{2}{e}^{-70/365}$

  $-\frac{{\left(70/365\right)}^3}{3!}{e}^{-70/365}-e^{-70/365}\frac{{\left(70/365\right)}^4}{4!}-e^{-70/365}\frac{{\left(70/365\right)}^5}{5!}$

  $=1-e^{-70/365}\left[1+\frac{70}{365}+\frac{{\left(70/365\right)}^2}{2!}+\mathrm{....}+\frac{{\left(70/365\right)}^2}{5!}\right]$

  $\begin{array}{l}=1-\left(0.8255\right)\left[1+0.1918+0.0184+0.001176+0.0000564+0.0000002\right]\\ =0\end{array}$

e

Summary Introduction

Interpretation:

Reason for low value in part (d)

Concept Introduction:

Probability is the likelihood of an event to occur.. It ranges between 0 to 1. O implies no chance of occurance while 1 implies 100% chance of occurance.

Explanation of Solution

The given period between Thanksgiving Day and New Year’s Eve is expected to be busier as compared to normal time period. Therefore, the probability for more than 5 breakdowns would be low during the period.