Question

Consider the 1981 Super Bowl commercial from Schlitz involving a live taste test (Links to an external site.). Suppose that a taste tester preferring Schlitz is considered a success which occurs with probability .5. In a sample of 50 what is the probability that 25 or more will choose Schlitz as the best beer?

Answer 1

Answer:

$P(X\geq 25) = 1-P(X<25) = 1-P(X\leq 24)=0.556$

See explanation below.

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we know that:

$X \sim Binom(n=50, p=0.5)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

And we want to find this probability:

$P(X\geq 25)$

And we can find this probability using the complement rule like this:

$P(X\geq 25) = 1-P(X<25) = 1-P(X\leq 24)= 1-[P(X=0)+P(X=1)+P(X=2)+.....+P(X=24)]$

And in order to do the operations we can use the following excel code:

"=1-BINOM.DIST(24,50,0.5,TRUE)"

And we got:

$P(X\geq 25) = 1-P(X<25) = 1-P(X\leq 24)=0.556$