We are given the results of a survey that investigated how people eat corn. It occurs that

80 %

of respondents — which accurately represent the population — eat corn on the cob in circles rather than from side to side. We want to find the probability that at least two out of five people eat corn on the cob in circles.

P (at least 2) = P (2, 3, 4, or 5)

We need to find the probabilities that particular numbers of people eat corn on the cob in circles. To do so, let's first check if the experiment is a binomial experiment.

Feature of a Binomial Experiment	Our Experiment	Does It Meet Requirements?
There are a fixed number of trials.	Five people are considered.	$✓$
Each trial has two possible outcomes.	Eating corn on the cob in circles or from side to side.	$✓$
The trials are independent.	The way one person eats corn on the cob does not affect the probability of a method chosen by the other people.	$✓$
The probability of each outcome is constant throughout the trials.	The probability that a person eats corn on the cob in circles is $80 % .$ The experiment does not affect the results of the survey.	$✓$

Once we know that the experiment is a binomial experiment, we can apply the Binomial Theorem to find the probabilities that a particular number of people eat corn on the cob in circles, which we will call a success.

Binomial Theorem
For every positive integer $n,$ $(a + b)^{n} =_{n} C_{0} a^{n} +_{n} C_{1} a^{n - 1} b +_{n} C_{2} a^{n - 2} b^{2} + \dots +_{n} C_{n - 1} a b^{n - 1} +_{n} C_{n} b^{n}$

Let $p$ and $q$ denote the probability of a success and a failure, respectively. The survey shows that $p = 0.8 .$ Knowing that $p + q = 1,$ we can find that $q = 1 - 0.8 = 0.2 .$ Since there are $five$ trials, we will expand the binomial $(p + q)^{5} .$ Recall that each term of the expansion represents the probability.

We will now substitute the given values to calculate each term of the expansion.

p^{5} + 5 p^{4} q + 10 p^{3} q^{2} + 10 p^{2} q^{3} + 5 p q^{4} + q^{5}

SubstituteII

$p = 0.8$ , $q = 0.2$

0.8^{5} + 5 (0.8)^{4} (0.2) + 10 (0.8)^{3} (0.2)^{2} + 10 (0.8)^{2} (0.2)^{3} + 5 (0.8) (0.2)^{4} + 0.2^{5}

CalcPow

Calculate power

0.32768 + 5 (0.4096) (0.2) + 10 (0.512) (0.04) + 10 (0.64) (0.008) + 5 (0.8) (0.0016) + 0.00032

Multiply

0.32768 + 0.4096 + 0.2048 + 0.0512 + 0.00064 + 0.00032

Recall that each term gives the probability of a particular number of successes in the experiment. Let's present the values in a table.

Number of People	Probability
$5$	$0.32768$
$4$	$0.4096$
$3$	$0.2048$
$2$	$0.0512$
$1$	$0.00064$
$0$	$0.00032$

We can also present the probabilities using a bar graph. It will show us the probability distribution of the number of successes.

We want to calculate the probability that at least

2

people eat corn on the cob in circles. To do so we need to sum up the probabilities for

5,

4,

3,

and

2

people.

P (2, 3, 4, or 5) = 0.32768 + 0.4096 + 0.2048 + 0.0512

AddTerms

Add terms

P (2, 3, 4, or 5) = 0.99328

Exercise

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