Question

a. For the experiment in which the number of computers in use at a six - computer lab is observed, let B, C be the events defined as B = {3, 4, 5, 6}, and C = {1, 3, 5}. Give the event (B ^ C) using set notation (i.e using { } ).

b. Suppose that the probability of a person getting a certain rare disease is 0.0004 . Consider a town of 10,000 people. What is the approximate probability of seeing more than 3 new cases in a year?

c. To get to work, a commuter must cross train tracks. The time the train arrives varies slightly from day to day, but the commuter estimates he will be stopped 10% of work days. During a certain 5 - day work week, what is the probability that he gets stopped at least once during the week?

d. Suppose occurrences of sales on a small company’s website are modeled by a Poisson model with λ = 6/hour. What is the probability that the next sale will happen in the next 12 minutes?

Answer 1

solution:

a) Given that

U = { 1,2,3,4,5,6}

B = {3,4,5,6} and C = {1,3,5}

B's complement - B^c = U - B =  { 1,2,3,4,5,6} - {3,4,5,6} = {1,2}

Note: sometimes logical notation - B conjunction C - BΛC = B and C ={3,5}

b) Given that

The probability of a person getting certain rare disease = 0.0004

Total No.of people = 10,000

Using poisson Distribution, we have

P(X=x) = (e^-λ * λ ^x) / x!

Here, λ = np [ since, poisson distribution is the limiting case of Binomial Distribution]

   λ = 10000 * 0.0004 = 4

P(seing more than 3 cases per year) = P(X>3)

    = 1 - P(X<=3)

= 1 - [ P(X=0) + P(X=1) + P(x=2) + P(X=3)]

= 1 - [ (e^-4 * 4 ^0) / 0! + (e^-4 * 4 ^1) / 1! + (e^-4 * 4 ^2) / 2! + (e^-4 * 4 ^3) / 3! ]

= 1 - [  e^-4 + 4 e^-4 + 8e^-4 + 10.7e^-4 ]

= 1 - 20.7e^-4

= 0.621

Probability of seeing more than 3 cases per year = 0.621

c) Given that

Probability that he will be stopped = 10% = 0.10

No.of working days in a certain week = 5

Probability that he never stops = 1 -   Probability that he will be stopped = 1 - 0.10 = 0.90

Probability that he gets stopped at least once during week) = 1 - P(never gets stopped)    [since,Using Complement rule ]

= 1 - (0.90)^5

= 0.41

d) Given that

λ = 6/ hour   

First,  Convert the 12 mins to hours = 12 min *(1 hr / 60 min) = 0.2

Therefore, t = 0.2

If the discrete random variable modeled by a poisson model with parameter  λ ,then the times between those events can be modeled by exponential distribution with same parameter  λ.

To find probability given by

P(X<=t) = 1 - e^-( λt) = 1 - e^(- 6 * 0.2) = 0.699 ~ 0.70

The probability that next sale will happen in the next 12 minutes = 0.70[approximately rounded to 2 decimals]