Chapter 3, Problem 1UA

a.

To determine

To find: The probability of winning the lottery next Tuesday and Wednesday.

Answer to Problem 1UA

The probability of winning the lottery next Tuesday and Wednesday is 0.000001.

Explanation of Solution

Given info:

A lottery has a three digit number from 000 to 999. Also, the number is 389.

Calculation:

Let event A denote the winning ticket on next Tuesday and B be denote that winning ticket on next Wednesday.

Here, the winning ticket on next Wednesday is not affected by winning ticket on next Tuesday. Thus, the selections are independent.

The formula for probability of event A is,

$P\left(A\right)=\frac{\text{Number of outcomes in event }A}{\text{Total number of outcomes in sample space}}$

Substitute 1 for ‘Number of outcomes in event A’ and 1,000 for ‘Total number of outcomes in sample space’,

$\begin{array}{c}P\left(A\right)=\frac{1}{1,000}\\ =0.001\end{array}$

The formula for probability of event B is,

$P\left(B\right)=\frac{\text{Number of outcomes in event }B}{\text{Total number of outcomes in sample space}}$

Substitute 1 for ‘Number of outcomes in event B’ and 1,000 for ‘Total number of outcomes in sample space’,

$\begin{array}{c}P\left(B\right)=\frac{1}{1,000}\\ =0.001\end{array}$

Multiplication rule for Independent Events:

If the events A and B are independent the probability of occurring of event B is does not affected by event A, that is $P\left(A\text{ and }B\right)$ becomes $P\left(A\right)\times P\left(B\right)$.

The required probability is,

$\begin{array}{c}P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)\\ =0.001\times 0.001\\ =0.000001\end{array}$

Therefore, the probability of winning next Tuesday and Wednesday is 0.000001.

Interpretation:

There is a 0.000001 probability of winning next Tuesday and Wednesday.

b.

To determine

To find: The probability of winning on Wednesday.

Answer to Problem 1UA

The probability of winning on Wednesday is 0.001.

Explanation of Solution

Given info:

Won on Tuesday.

Calculation:

Let event A denote the winning ticket on Tuesday and B be denote that winning ticket on Wednesday.

Here, the winning ticket on next Wednesday is not affected by winning ticket on next Tuesday. Thus, the selections are independent.

The formula for probability of event B is,

$P\left(B\right)=\frac{\text{Number of outcomes in event }B}{\text{Total number of outcomes in sample space}}$

Substitute 1 for ‘Number of outcomes in event B’ and 1,000 for ‘Total number of outcomes in sample space’,

$\begin{array}{c}P\left(B\right)=\frac{1}{1,000}\\ =0.001\end{array}$

Thus, there is a 0.001 probability of winning on Wednesday given that you won on Tues day.

Interpretation:

There is a 0.001 probability of winning on Wednesday given that you won on Tues day.

c.

To determine

To find: The probability of winning on Wednesday.

Answer to Problem 1UA

The probability of winning on Wednesday is 0.001.

Explanation of Solution

Given info:

Did not win on Tuesday.

Calculation:

Let event A denote the winning ticket on Tuesday and B be denote that winning ticket on Wednesday.

Here, the winning ticket on next Wednesday is not affected by winning ticket on next Tuesday. Thus, the selections are independent.

The formula for probability of event B is,

$P\left(B\right)=\frac{\text{Number of outcomes in event }B}{\text{Total number of outcomes in sample space}}$

Substitute 1 for ‘Number of outcomes in event B’ and 1,000 for ‘Total number of outcomes in sample space’,

$\begin{array}{c}P\left(B\right)=\frac{1}{1,000}\\ =0.001\end{array}$

Thus, there is a 0.001 probability of winning on Wednesday given that you did not win on Tuesday.

Interpretation:

There is a 0.001 probability of winning on Wednesday given that you did not win on Tuesday.