Probability
Home

W04 - HW Solutions

01

(1) Consider the total number of outcomes.

Since there are two dice being rolled, there are 36 total outcomes.

(2) Consider the total number of desired outcomes for .

There are 3 total desired outcomes, , , (6, 4).

(3) Use formula to find .

(4) Consider the total number of desired outcomes for .

There is only 1 desired outcome, .

(5) Use formula to find .

(6) Consider the total number of desired outcomes for .

There are 6 total desired outcomes: , , , , ,

Use formula to find .

02

(1) Identify distribution.

We have that , since there are trials and .

(2) Use binomial distribution formula to get PMF.

(3) Find CDF.

Since , we have

(4) Write out explicit values of CDF.

03

04

(1) Define random variables.

Let .

We wish to find .

(2) Compute using the formula for a geometric distribution.

05

(1) Define random variables.

Let represent the car with three components.

Let represent the car with five components.

(2) Find probabilities that both cars work.

For the three-component car, we want , so

For the five-component car, we want , so

(3) Find when .

Solve the inequality for .

06

(1) Set up conditional probability formula.

(2) Find formulas for numerator and denominator.

(3) Plug in values into initial formula

07

(1) Find formula for ratio .

(2) Interpret ratio.

We want , so .

Solving for , we get .

Since is an integer, the is maximized when .

(3) Compute directly.

Based on the figure, and .

(4) Use ratio to solve for successive terms.

(5) Add up probabilities.

08

(a)

Identify the distribution.

follows a negative binomial distribution with parameters and .

(b)

Compute .

(c)

Compute .

We know that the minimum number of passersby before a winner is declared is .

Therefore, .

09

(1) Compute .

There are total students.

Let be the probability that Bus is selected, where .

Note that .

(2) Compute .

Let be the probability that Bus is selected.

Since it’s based off the drivers, for all .

(3) Interpret solution.

, because the probability that bus was selected in both scenarios varied.

10

(a)

(b)

11

(1) State the PMF of a geometric random variable.

(2) Use formula for expectation to find .

(3) Apply hint.

We have that .

Differentiating both sides yields .

Note that here, , so

(4) Find expression , and note that .

Applying the hint, we have .

Using the linearity of expectation, we can write this as .

(5) Find and

First, note that the second derivative of is .

Thus, .

(6) Find .

12

(i):

(ii):

So event (i) is more probable.

13

(a)

(b)