Question

Question:In a super lottery, a player wins a fortune if they choose the eight numbers selected by a computer from the positive integers not exceeding 100. What is the probability that a player wins this super lottery?

Short answer

Answer

The probability that a person wins the grand prize by picking 7 numbers that are among the 11 numbers selected at random by a computer, provided a player selects 7 numbers out of the first 80 positive integers is $P\left(E\right)=5.37\times {10}^{-12}$.

Step-by-step solution

 Given  

The positive integers not exceeding 100.

The Concept of Probability

If$\mathit{S}$represents the sample space and$\mathit{E}$ represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$\mathit{P}\mathbf{\left(}\mathit{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{n}\mathbf{\left(}\mathbf{E}\mathbf{\right)}}{\mathbf{n}\mathbf{\left(}\mathbf{S}\mathbf{\right)}}$

Determine the probability

As per the problem we have been asked to find the probability that a person wins the grand prize by picking 8 numbers selected by a computer from the positive integers not exceeding 100.

If $S$represents the sample space and $E$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$.

So, the number of possible ways of select 8 numbers out of the first 100 positive integers is$n\left(s\right){=}^{100}{C}_8$.

Total number of ways that the 8 numbers selected by a player to be in 8 numbers selected by computer at random is$n\left(E\right){=}^8{C}_8$.

There will be only one set of positive integers for which lottery can be drawn.

The probability that a person wins the grand prize by picking 8 numbers selected by a computer from the positive integers not exceeding 100 is as follows:

$$\begin{gathered} P\left(E\right)=\frac{{}^8{C}_8}{{}^{100}{C}_8} \\ P\left(E\right)=\frac{1}{186087894300} \\ P\left(E\right)=5.37\times {10}^{-12} \end{gathered}$$