Question

Consider three classes, each consisting of 20 students. From this group of 60 students, a group of 3 students is to be chosen. i. What is the probability that all 3 students are in the same class? [3 marks] ii. What is the probability that 2 of the 3 students are in the same class and the other student is in a different class? [3 marks] (b) In a new casino game, two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls to see if you will win any money. Suppose that you win $2 for each black ball selected, you lose $1 for each white ball selected, and you get nothing for each orange ball selected. If the casino lets you play this new casino game with no entry fee, what is the probability that you will not lose any money? [7 marks]

(c) A survey was conducted on lawyers at two different law firms about their annual incomes. The following table displays data for the 275 lawyers who responded to the survey. Annual Income Law Firm 1 Law Firm 2 Total Under $45,000 30 20 50 $45,000 to $89,999 35 40 75 $90,000 and over 100 50 150 TOTAL 165 110 275 Suppose we choose a random lawyer who responded to the survey. Are the events ”income is under $45,000” and ”employed at Law Firm 2” independent? [4 marks] (d) A smartphone company receives shipments of smartphones from three factories, labelled, 1, 2 and 3. Twenty-five percent of shipments come from factory 1 whose shipments contain 8% defective smartphones. Sixty-five percent of the shipments come from factory 2 whose shipments contain 6% defective smartphones. The remainder of the shipments comes from factory 3 whose shipments contain 4% smartphones. The company receives a shipment, but does not know the source. A random sample of 15 smartphones is inspected, and three of the smartphones are found to be defective. What is the probability that this shipment came from factory 2?

Answer 1

(a)

(i)

From 60 students 3 students can be choose in ways

To count number of ways to choose 3 students from the same class,

• number of ways to choose a class is
• number of ways to choose 3 students from a class is

So, number of ways to choose 3 students from the same class is

Hence, required probability is given by

(ii)

From 60 students 3 students can be choose in ways

Number of ways to choose 2 students from a class and one from other class is

Hence, required probability is given by

(b)

This problem can be solved by using complement probability. So we first find the probability of losing money.

Money is lost if

• both drawn balls are white
• one ball is white and one is orange

Possible ways to choose 2 balls from (8+4+2)=14 balls is

Possible ways to choose 2 balls from 8 white balls is

Possible ways to choose 1 ball from 8 white balls and 1 ball from 2 orange balls is

So, number of ways in which money is lost =

So, probability of losing money is given by

Hence, probability that we shall not lose money

(c)

Two events are said to be independent if cell frequency corresponding to those events is product of corresponding row total frequency and corresponding column total frequency divided by total frequency.

Thus the events 'income is under $45000' and 'employed at Law Firm 2' are independent if

where

frequency of the cell in the intersection in 1 st row and 2 nd column

= row total of first row

= column total of second column

= total frequency

Thus we get,

We observe that,

Hence, the events 'income is under $45000' and 'employed at Law Firm 2' are independent.