Chapter 11.9, Problem 25PPSE

Solution

To calculate: The probability that at least two out of five people you know eat corn on the cob in circles.

The probability that at least two out of five people you know eat corn on the cob in circles is 0.9933.

Given:

The percentage of respondents who eat corn on the cob in circles is $=80\%$

The total number of respondents selected $n=5$

Concept used:

The probability with respect to a binomial probability distribution for x number of success with parameters $(n,p)$ where n is the total number of trials and p is the probability of success is calculated using the formula $P(X=x)=\frac{n!}{x!\left(n-x\right)!}{p}^x{(1-p)}^{n-x}$

Calculation:

There are about 80% of respondents eat corn on the cob in circles, thus the probability that a randomly selected respondent eats corn on the cob in circles is $p=\frac{80}{100}=0.80$ .

The probability that a randomly selected respondent does not eat corn on the cob in circles is $1-p=1-0.8=0.2$

The total number of respondents selected $n=5$

The probability that there are at least two out of five respondents eat corn on the cob in circles is calculated as shown below

At least two out of five people have to eat corn on the cob in circles, thus it can be two or three or four or five people who eat the corn on the cob in circles.

X is a binomial distribution representing the number of people who eat corn on the cob in circles with parameters $(n=5,p=0.8)$

  $\begin{array}{}$ P(X=2)=\frac{5!}{2!\left(5-2\right)!}{(0.8)}^2{(1-0.8)}^{5-2}$$ P(X=2)=10\times {(0.8)}^2{(0.2)}^3$$ P(X=2)=0.0512$$ $\end{array}$

  $\begin{array}{}$ P(X=3)=\frac{5!}{3!\left(5-3\right)!}{(0.8)}^3{(1-0.8)}^{5-3}$$ P(X=3)=10\times {(0.8)}^3{(0.2)}^2$$ P(X=3)=0.2048$$ P(X=4)=\frac{5!}{4!\left(5-4\right)!}{(0.8)}^4{(1-0.8)}^{5-4}$$ P(X=4)=5\times {(0.8)}^4{(0.2)}^1$$ P(X=4)=0.4096$$ P(X=5)=\frac{5!}{5!\left(5-5\right)!}{(0.8)}^5{(1-0.8)}^{5-5}$$ P(X=5)=1\times {(0.8)}^5$$ P(X=5)=0.3277$\end{array}$

The probability that there are at least two out of five people eat the corn on the cob in circles is $=0.0512+0.2048+0.4096+0.3277=0.9933$ (rounded to 4 decimal places).

Interpretation:

When the percentage of respondents eat corn on the cob in circles is known probability can be calculated by writing the percentage in decimal form by dividing it by 100 and the probability that one person eats corn on the cob in circles is calculated as 0.8. A binomial probability distribution is used to determine probability for number of success among fixed n number of trails where each independent trail has two possible outcomes which are treated as success or failure. Here a person eats corn on the cob in circles or not are two possible outcomes and there are a total of five persons. Thus binomial probability distribution helps to calculate the probability that two or more people eats corn on the cob in circles among five people.