FPM Term 1: 2015-2016 Probability 04/09/2015

1

Assignment 1

Due Date: September 10, 2015

Topics covered: Sample Space, Probability, Conditional Probability, and Independence

1. A group of five strangers start to climb the stairs in their apartment building from the ground floor. There

are 10 floors above.

a. Specify an appropriate sample space and determine the probability that they are all going to

different floors.

b. How does the answer change when each person chooses with probability 1/2 the 10th floor as

the exit floor and the other floors are equally likely as the exit floor?

Mention any assumptions you have made.

2. Maya tosses ‘r + 1’ fair coins and Chaya tosses ‘r’ fair coins. What is the probability that Maya gets more

tails than Chaya?

3. A machine produces components for use in cellular phones. At any given time, the machine may be in

one, and only one, of three states: operational, out of control, or down. From experience with this

machine, a quality control engineer knows that the probability that the machine is out of control at any

moment is 0.02, and the probability that it is down is 0.015.

a. What is the relationship between the two events “machine is out of control” and “machine is

down”?

b. When the machine is either out of control or down, a repair person must be called. What is the

probability that a repair person must be called right now?

c. Unless the machine is down, it can be used to produce a single item. What is the probability that

the machine can be used to produce a single component right now? What is the relationship

between this event and the event “machine is down”?

4. When one is randomly sampling four items from a large number of elements, say ‘n’, what is the

probability that all four elements will come from the top quartile of the distribution? What is the

probability that at least one of the four elements will come from the bottom quartile of the distribution?

**

5. Two chefs A and B are challenged to separately create a new dish in 2 hours. From past experience we

know that:

a. The probability that chef A’s dish is a hit is 2/3

b. The probability that chef B’s dish is a hit is 1/2

c. The probability that at least one of their dishes is a hit is 3/4

Assuming that only one of the dishes can be labeled a hit, what is the probability that it was created by

chef A?

FPM Term 1: 2015-2016 Probability 04/09/2015

2

6. I am taking a probability class and in each week, I can be either up-to-date or I may have fallen behind. If I

am up-to-date in a given week, the probability that I will be up-to-date the following week is 0.8. If I am

behind in the given week, the probability that I will be up-to-date the following week is 0.4. Given that I

am up-to-date this week, what is the probability that I will be up-to-date three weeks later?

7. Prove that if A and B are independent so are

a. A and BC

b. Ac and B

c

8. Suppose that A and B are events such that P(A|B) = P(B|A) and P(A∪B) = 1 and P(A ∩ B) > 0. Prove that

P(A) > 1/2.

** Question 4:

Quartiles break down the distribution of the data (the way the data points are distributed along the number line)

into four groups i.e. first quarter, second quarter, third quarter, and fourth quarter. In the question, by 1st

quartile

we mean those data points in the first quarter.