Question

4 shoppers are asked to state their preference for 2 brands Coke or Pepsi Let A denote the event that exactly two of the four individuals will prefer Coke. List the elements in A and find P(A)=?

Answer 1

`A = {< C,C,P,P >, < C,P,C,P>, < C,P,P,C >, < P,C,C,P >,< P,C,P,C >, < P,P,C,C> }`
`P(A)=6/16=3/8`

Explanation:

Explanation of notation:
The ordered sequence: `< color(red)C,color(blue)P,color(green)P,color(magenta)C >`
Is intended to indicate that
`color(white)("XXX")`the first person chose `color(red)C`oke;
`color(white)("XXX")`the second person chose `color(blue)P`epsi;
`color(white)("XXX")`the third person chose `color(green)P`epsi; and
`color(white)("XXX")`the fourth person chose `color(magenta)C`oke.

With 4 shopper each of whom can choose 1 of 2 drinks,
there are `2^4=16` possible outcome sequences.

Either from the listed set of elements of `A` (above) or by recognizing that there are `""_4C_2=(4!)/(2!2!)=6` outcomes with 2 out of the 4 selections being Pepsi,
we see that the probability of exactly 2 shopper selecting Pepsi is
`color(white)("XXX")P(A)=6/16=3/8`

Warning
An assumption was required that a randomly selected shopper would be equally likely to chose Coke or Pepsi. If the population was biased in favor of one or the other this analysis would be invalid. (As an extreme example, for demonstration purposes: suppose everyone loved Pepsi and hated Coke; the the only outcome that would ever occur would be `< P,P,P,P >` and `P(A)=0`.)