Chapter 13, Problem 15CT

To determine

To calculate: The probability of winning a lottery, if in a lottery, there are ten balls numbered $1,2,3,4,5,6,7,8,9,10$. Of these, five are drawn in order. If you pick five numbers that match those drawn in the correct order, you win $\$1,000,000$.

Answer to Problem 15CT

Solution:

The probability of winning a lottery is $0.000033$.

Explanation of Solution

Given information:

In a lottery, there are ten balls numbered $1,2,3,4,5,6,7,8,9,10$. Of these, five are drawn in order. If you pick five numbers that match those drawn in the correct order, you win $\$1,000,000$.

Formula used:

The formula to find probability for any event $E$ is;

$P\left(E\right)=\frac{\text{Number}\text{of}\text{ways}\text{that}\text{E}\text{can}\text{occur}}{\text{Number}\text{of}\text{possible}\text{outcomes}}$.

The number of ways of selecting $r$ objects from $n$ distinct objects, $r\le n$, in which repetition of objects is not allowed and order is important is given by the formula,

$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

In a lottery, there are ten balls numbered $1,2,3,4,5,6,7,8,9,10$. Of these, five are drawn in order.

And, win $\$1,000,000$ if picked five numbers matches those drawn in the correct order, So, to calculate the probability of winning a lottery, first calculate total number of ways to pick five numbers from given $10$ numbers in order.

To select numbers order is important hence use permutation to find total number of ways to pick five numbers from given $10$ numbers.

Here, $n=10$ and $r=5$.

Thus, $P\left(10,5\right)=\frac{10!}{\left(10-5\right)!}=\frac{10!}{5!}$.

By writing $10!$ in factorial notation,

$\frac{10!}{5!}=\frac{10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5!}{5!}$.

$\Rightarrow P\left(10,5\right)=10\cdot 9\cdot 8\cdot 7\cdot 6=30240$.

So here total number of ways to pick fives number is $30240$.

Out of $30240$, only one order is match to winning order.

Thus, $probabilityofwinning=\frac{1}{30240}=0.000033$.

Therefore, the probability of winning a lottery is $0.000033$.