Question

New Jersey Lottery Let A denote the event of placing a $1 straight bet on the New Jersey Pick 3 lottery and winning. There are 1000 different ways that you can select the three digits (with repetition allowed) in this lottery, and only one of those three-digit numbers will be the winner. What is the value of? What is the value of$P\left(\overline{A}\right)$?

Short answer

The value of $P\left(A\right)$is equal to $\frac{1}{1000}=0.001$.

The value of $P\left(\overline{A}\right)$is equal to $\frac{999}{1000}=0.999$.

Step-by-step solution

Given information

A is the event of winning a bet of $1 on the New Jersey Pick 3 lottery. The lottery involves choosing three digits (with repetition). Only one combination of the chosen three digits will result in winning.

Define the probability of an event and its complement

Theprobability of an event is defined as the chance of occurrence of that event out of all the possible outcomes.

For any arbitrary event, the probability of occurrence of E is computed as shown below:

$P\left(E\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}}$

Let $\overline{E}$be the outcome of non-occurrence of E (complementary event of E). Then, the probability of non-occurrence of E is given below:

$P\left(\overline{E}\right)=1-P\left(E\right)$

Calculation of PA

The number of ways of winning the lottery is 1.

The total number of ways of choosing three digits is 1000.

The probability of winning the lottery$P\left(A\right)$ is obtained as follows:

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}} \\ =\frac{1}{1000} \\ =0.001 \end{gathered}$$

Therefore, $P\left(A\right)$is equal to 0.001.

Calculation of PA¯

The probability of occurrence for the complement of event A is computed as follows:

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-\frac{1}{1000} \\ =\frac{999}{1000} \\ =0.999 \end{gathered}$$

Therefore, is equal to 0.999.