Chapter 10.5, Problem 49E

To determine

To calculate:

The probability of winning a lottery.

Answer to Problem 49E

The probability of winning a lottery is $\frac{1}{15890700}$ .

Explanation of Solution

Given information:

Selected numbers $=6$

Each an integer from $=0to49$

Formula used:

Expression for combination

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!\cdot r!}$

Where,

  $n=$ objects

  $r=$ time

Calculation:

First, write the number of possible outcomes as the number of combinations of $50$ numbers taken $6$ at a time, ${}_{50}{C}_6$ because the order is not important.

So, apply the expression for combination for $n$ objects taken $r$ at a time is as follows:

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!\cdot r!}$

The required number of combinations are as follows:

  $\begin{array}{l}{}_{50}{C}_6=\frac{50!}{\left(50-6\right)!\cdot 6!}\\ =\frac{50!}{44!\cdot 6!}\\ =\frac{50\cdot 49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44!}{44!\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\\ =15890700\end{array}$

Now calculate the probability of favorable outcomes. So, the probability is as follows:

  $P\left(winninglottery\right)=\frac{1}{15890700}$

Hence, the probability of winning a lottery is $\frac{1}{15890700}$ .