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Q31E

Expert-verifiedFound in: Page 451

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**Question:** **Suppose that 100 people enter a contest and that different winners are selected at random for first, second, and third prizes. What is the probability that Michelle wins one of these prizes if she is one of the contestants?**

**Answer **

The probability that Michelle wins one of the three prizes if she is one of the 100 contestants is$P\left(E\right)=0.03$ .

Michelle being one of the 100 contestants

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:

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

** **

As per the problem we have been asked to find the probability that Michelle wins one of the three prizes if she is one of the 100 contestants taking part in a contest and different winners are selected at random for first, second, third prizes.

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:

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

The total number of ways in which any three random people wins $n\left(S\right){=}^{100}{C}_{3}$.

Number of ways in which three people win among which Michelle wins one are $n\left(E\right){=}^{99}{C}_{2}$.

Now substitute the values in the above formula we get,

$P\left(E\right)=\frac{{}^{99}{C}_{2}}{{}^{100}{C}_{3}}\phantom{\rule{0ex}{0ex}}P\left(E\right)=\frac{99\times 98}{2}\times \frac{3\times 2}{100\times 99\times 98}\phantom{\rule{0ex}{0ex}}P\left(E\right)=\frac{3}{100}\phantom{\rule{0ex}{0ex}}=0.03\phantom{\rule{0ex}{0ex}}$

The probability that Michelle wins one of the three prizes if she is one of the 100 contestants is$P\left(E\right)=0.03$ .

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