Question

2. Of all people who enter Rooster’s Coffeehouse, 79% order a beverage, 53% order food, and 45% orderboth a beverage and food.

(a) You plan to select one customer at random. Explicitly define the two events that could occur. Express each of the three probabilities listed above as the probability in terms of these two events.

Note: For each part in the remainder of this question, you must first express each probability in terms of the events you defined in part (a) and justify any computation through the use of a formula.

(b) Are ordering a beverage and ordering food mutually exclusive events? Explain using only one of the probabilities you stated in part (a).

(c) What is the probability that a customer does not order food?

(d) What proportion of all people ordered a beverage, food, or both?

(e) What proportion of people order no beverage and no food? (HINT: How does this relate to part (d)?) erage and no food? (HINT: How does this relate to part (d)?)

Answer 1

Solution:

Given: Of all people who enter Rooster’s Coffeehouse, 79% order a beverage, 53% order food, and 45% order both a beverage and food.

Part a)

Let B = customer who enter Rooster’s Coffeehouse order a Beverage and

let F = customer who enter Rooster’s Coffeehouse order a Food.

Thus we have:

P(B) = P( customer who enter Rooster’s Coffeehouse order a Beverage)

P(B) = 79%

P(B) = 0.79

P(F) = P(  customer who enter Rooster’s Coffeehouse order a Food)

P(F) = 53%

P(F) = 0.53

and

P( customer who enter Rooster’s Coffeehouse order both a beverage and food)

45%

0.45

Part b) Are ordering a beverage and ordering food mutually exclusive events?

No , since   0.45 which is not equal to 0.

Part c) What is the probability that a customer does not order food?

If F =  customer who enter Rooster’s Coffeehouse order a Food,

then F^c =   customer who enter Rooster’s Coffeehouse does not order a Food,

P( F^c ) = 1 - P( F )

P( F^c ) = 1 - 0.53

P( F^c ) = 0.47

Thus the probability that a customer does not order food is 0.47

Part d) What proportion of all people ordered a beverage, food, or both?

That is find:

Using addition rule of probability, we get:

Thus proportion of all people ordered a beverage, food, or both is 0.87.

Part e) What proportion of people order no beverage and no food?

That is find:

Using De'Morgans law we get:

From part d) we have:

Thus