a We want to find the probability that exactly 12 out of 20 people are in favor of a new environmental law.
To find it, since this probability experiment satisfies the properties of a binomial experiment, we can use the Binomial Probability Formula. Let's first define the success and the failure in addition to their probabilities.
Success
Failure
Event
In favor of the new environmental law
Against the new environmental law
Probability
0.5
1- 0.5= 0.5
Now, let's recall the Binomial Probability Formula before using it.
Binomial Probability Formula
The probability of x successes in n binomial trials, where p and q are the probabilities of success and failure, respectively, can be calculated with the following formula.
P( x)= _nC_x p^x q^(n- x)
We will substitute values into the formula.
P( x)= _nC_x p^x q^(n- x)
⇓
P( 12)= _(20)C_(12)( 0.5)^(12)( 0.5)^(20- 12)
From here we need to calculate _nC_x. To do so, we will remember the Combination Formula.
_nC_x =n!/x!( n- x)!
Therefore, we can substitute 20! 12!( 20- 12)! for _(20)C_(12).
The probability of exactly 12 successes in 20 trials is about 0.1201, or 12.01 %.
b We know from Part A that our situation is a binomial experiment.
With this information we will find the expected number of people in favor of the law. To do so, we will find the mean of the binomial distribution. Recall the mean of a binomial distribution.
Mean of a Binomial Distribution
The mean μ of a binomial distribution is given by μ=n* p, where n is the number of trials and p is the probability of success.
We know that p= 0.5 and n= 20. To evaluate the mean we will substitute these values into the equation.
We found that the mean is 10. It means that since 20 people took the survey and the probability that a person is in favor of the law is 0.5, the expected number of people in favor of the law is 10.
