Chapter 12.7, Problem 2CYP

To determine

To find the probability that at least $2$ of her game cards are winners.

Answer to Problem 2CYP

  $0.08146$ .

Explanation of Solution

Given:

Becky bought $5$ game cards at the store. Each game card has a $10\%$ chance of winning.

Calculation:

The probability of winning (success) is

  $p=\frac{10}{100}=\frac{1}{10}$

The probability of (failure) is

  $q=\frac{9}{10}$

The probability that at least $2$ of her game cards are winners.

  $\begin{array}{l}$ P(x)=n{C}_2{(p)}^x{q}^{n-x}$P(x\le 2)=C(5,2){\left(\frac{1}{10}\right)}^2{\left(\frac{9}{10}\right)}^3+C(5,3){\left(\frac{1}{10}\right)}^3{\left(\frac{9}{10}\right)}^2+C(5,4){\left(\frac{1}{10}\right)}^4{\left(\frac{9}{10}\right)}^1+C(5,5){\left(\frac{1}{10}\right)}^5{\left(\frac{9}{10}\right)}^0\\ P(x\le 2)=10\times \frac{729}{100000}+10\times \frac{81}{100000}+5\times \frac{9}{100000}+\frac{1}{100000}\\ =\frac{1}{100000}\left(7290+810+45+1\right)\\ =\frac{8146}{100000}=0.08146\end{array}$