Question

7. The daily special at a particular restaurant is a three-course meal consisting of an entree, a starch, and a dessert. The choices of entree are: Chicken or roast beef; the choices of starch are: Pasta or rice or potatoes; and, the choices of desert are: Ice cream or Jello or apple pie or a peach. Consider the experiment consisting of the selection of a meal, that is, the selection of one entree, one starch, and one dessert. Hint: Use the inclusion–exclusion method to determine the number of outcomes in the unions.

a) How many outcomes (meals) are in the sample space?

b) Let A be the event that ice cream is chosen. How many outcomes (meals) are in A?

c) Let B be the event that chicken is chosen. How many outcomes (meals) are in B?

d) Describe the event AB = A ∩B in words and list all the outcomes (meals) in the event.

e) Describe the event A ∪ B in words. How many outcomes are in A ∪ B?

f) Let C be the event that rice is chosen. How many outcomes are in C?

g) Describe the event ABC = A ∩ B ∩ C in words and list all the outcomes (meals) in the event.

h) Describe the event A ∪ B ∪ C in words. How many outcomes are in A ∪ B ∪ C?

Answer 1

Part a)

Solution:

We are given

Items for entree = 2

Items for starch = 3

Items for desert = 4

Total number of outcomes in sample space = 2*3*4 = 24

Part b)

Let A be the event that ice cream is chosen. Then we have

Items for entree = 2

Items for starch = 3

Items for desert = 1 (Here, we can only choose ice cream.)

Total number of outcomes in A = 2*3*1 = 6

Part c)

Let B be the event that chicken is chosen. Then we have

Items for entree = 1 (Here, we can only choose chicken.)

Items for starch = 3

Items for desert = 4

Total number of outcomes in B = 1*3*4 = 12

Part d)

AB = A∩B = it is the event that ice cream and chicken is chosen.

List of all outcomes for this event is given as below:

(Chicken, Pasta, Ice cream),

(Chicken, Rice, Ice cream),

(Chicken, potatoes, Ice cream)

Part e

A UB = It is the event that ice cream or chicken is chosen.

We are given

Items for entree = 2

Items for starch = 3

Items for desert = 4

(There is no any repeated item in A, and B. All items are different.)

Total number of items in AUB = 2+3+4 = 9

Part f

C = event that rice is chosen

Items for entree = 2

Items for starch = 1 (here, we can only chose rice)

Items for desert = 4

Total number of outcomes in C = 2*1*4 = 8

Part g

A∩B∩C = It is event that contains ice cream, chicken, and rice.

There is only one outcome in the event which is given as (Chicken, Rice, Ice cream).

Part h

We are given

Items for entree = 2

Items for starch = 3

Items for desert = 4

(There is no any repeated item in A, B, and C. All items are different.)

Total number of items in AUBUC = 2+3+4 = 9