Chapter 7, Problem 1RQ

1. Define the probability of an event when all outcomes are equally likely.

Mat is the probability that you select the six winning numbers in a lottery if the six different winning numbers are selected from the first 50 positive integers?

To determine

(a)

The probability of an event in a sample space when all the events are likely to occur with equal probability

Answer to Problem 1RQ

$P(E)=\frac{1}{n}$, where $E$ is any event in the sample space. (1)

Explanation of Solution

Given:

A sample space with n equally likely events (outcomes)

If n events are there with equal probabilities, then

${\displaystyle \sum _1^{n}P(E_i})=1$ (2)

As all the terms in (2) are equal, it follows that

$P(E)=\frac{1}{n}$.

So (1) follows.

To determine

(b)

The probability of the occurrence of $6$ equally likely events out of $50$.

Answer to Problem 1RQ

$\frac{6}{50}=\frac{3}{25}$

Explanation of Solution

Given:

Sample space consisting of $\{1,2,3,\mathrm{...},50\}$ with equally likely events

Calculation:

Usinga), we have $P(1)=P(2)=\mathrm{..}=P(50)=\frac{1}{50}$

So, $P(\{1,2,3,4,5,6\})=\frac{1}{50}+\frac{1}{50}+\frac{1}{50}+\frac{1}{50}+\frac{1}{50}+\frac{1}{50}=\frac{6}{50}=\frac{3}{25}$