Question

Suppose that 55% of all adults regularly consume coffee, 45% regularly consume carbonated soda, and 70% regularly consume at least one of these two types of drinks.

(a) What is the probability that a randomly selected adult regularly consumes both coffee and soda?
(b) What is the probability that a randomly selected adult doesn't regularly consume at least one of these two products?
(c) What is the probability that a randomly selected adult regularly consumes coffee but does not regularly consume soda?

Answer 1

a) The probability that a randomly selected adult consumes both cofee and soda is 0.3

b) the probability that a randomly selected adult doesnt consume at least one of cofee and soda is 0.7

c) The probability that a randomly selected adult consumes cofee but not soda is 0.25

Explanation:

Lets pick a random adult and use the random variables

C = the adult consumes cofee

S = the adult consumes sodaa

we have that

P(C) = 0.55

P(S) = 0.45

P(C U S) = 0.7

a) We know that

0.7 = P(C U S) = P(C) + P(S) - P(C ∩ S) = 0.55 + 0.45 - P(C ∩ S)

Therefore

P(C ∩ S) = 0.55+0.45-0.7 = 0.3.

and as a consecuence, the probability that a randomly selected adult consumes both cofee and soda is 0.3.

b) The event 'a randomly selected adult doesnt consume at least one of the 2 products' is the complementary event of 'the adult consumes both cofee and soda', thus, the probability of this event is 1-P(C ∩ S) = 1 - 0.3 = 0.7.

c) Remember that

0.55 = P(C) = P(C ∩S) + P(C ∩ S^c) = 0.3 + P(C ∩ S^c)

Where S^c means that the adult doesnt consume soda. We can conclude that

P(C ∩ S^c) = 0.55-0.3 = 0.25

The probability that a randomly selected adult consumes cofee but not soda is 0.25.