Question

2) A travel agency is offering a special promotion for a 6-day holiday in Hawaii. They decided to do direct selling via phone to 400 families. The responses from the families are as follows: Income < $60,000 Income > $60,000 Booked the holiday 40 30 Did not book the holiday 210 120 Answer the following questions: (1 point each, except the first item, 2 points) 2a) Develop a joint probability table and show the marginal probabilities. 2b) What is the probability of a family whose income exceeds $60,000 and did not book the holiday? 2c) If income is < $60,000, what is the probability the holiday will be booked? 2d) If the holiday is booked, what is the probability that income exceeds $60,000?

Answer 1

2a)

Given,

	Income < $60,000	Income > $60,000
Booked the holiday	40	30
Did not book the holiday	210	120

Number of families the travel agency did direct selling via phone = 400

Joint probability table is obtained by dividing each number in the cell divided by 400;

	Income < $60,000	Income > $60,000
Booked the holiday	40/400=0.01	30/400=0.075
Did not book the holiday	210/400=0.525	120/400=0.3

The marginal probabilities are then obtained by columns sums and row sums

	Income < $60,000	Income > $60,000	Marginal Probability
Booked the holiday	0.1	0.075	0.1+0.075=0.175
Did not book the holiday	0.525	0.3	0.525+0.3 = 0.825
Marginal Probability	0.1+0.525=0.625	0.075+0.3=0.375

Joint Probability and the marginal probabilities

	Income < $60,000	Income > $60,000	Marginal Probability
Booked the holiday	0.1	0.075	0.175
Did not book the holiday	0.525	0.3	0.825
Marginal Probability	0.625	0.375

2b) Probability of a family whose income exceeds $6000 and did not take book the holiday = cell value against column : Income > $60,000 and row : Did not book the holiday

From the table , cell value against column : Income > $60,000 and row : Did not book the holiday = 0.3

Probability of a family whose income exceeds $6000 and did not take book the holiday = 0.3

2c)

If Income is < $60,000 , Probability that the holiday will be booked

= Probability that the holiday will be booked and Income is < $60,000 /  Probability that the Income is < $60,000

From the table,

Probability that the Income is < $60,000 = 0.625 (Marginal probability against  Income < $60,000)

Probability that the holiday will be booked and Income is < $60,000 = 0.1

If Income is < $60,000 , Probability that the holiday will be booked

= Probability that the holiday will be booked and Income is < $60,000 /  Probability that the Income is < $60,000

= 0.1 / 0.625 = 0.16

If Income is < $60,000 , Probability that the holiday will be booked = 0.16

2d)

If the holiday is booked, Probability that income exceeds $60,000

= Probability that income exceeds $60,000 and the holiday is booked / Probability of booked the holiday

Probability that income exceeds $60,000 and the holiday is booked = 0.075

Probability of booked the holiday = 0.175 ((Marginal probability against booked the holiday)

If the holiday is booked, Probability that income exceeds $60,000

= Probability that income exceeds $60,000 and the holiday is booked / Probability of booked the holiday

=0.075/0.175=0.428571429

If the holiday is booked, Probability that income exceeds $60,000 = 0.428571429