10.5. Practice Problems#

import math

10.5.1. Basic Ideas of Probability#

Further Reading: §2.1 in Navidi (2015)

10.5.1.1. Practice Problem 2.1-A#

Reference: §2.1, Problem 2 in Navidi.

A die (six faces) has the number 1 painted on three of its faces, the number 2 painted on two of its faces, and the number 3 painted on one face. Assume that each number is equally likely to come up.

  1. Find a sample space for this experiment

  2. Find \(P\)(odd number).

  3. If the die were loaded so that the face with the 3 on it were twice as likely to come up as each of the other five faces, would this chance the sample space? Explain.

  4. If the die were loaded so that the face with the 3 on it were twice as likely to come up as each of the other five faces, would this change the value of \(P\)(odd number)? Explain.

10.5.1.2. Practice Problem 2.1-B#

Adapted from §2.1, Problem 12 in Navidi.

Let \(V\) be the event that a computer contains a virus, and let \(W\) be the event that a computer contains a worm. Support \(P(V) = 0.15\), \(P(W) = 0.05\), and \(P(V \cup W) = 0.17\). Do the following:

  1. Draw a Venn diagram. Label \(V\), \(V^c\), \(W\), \(W^c\), \(V \cup W\), and \(V \cap W\).

  2. Find the probability that the computer contains both a virus and a worm.

  3. Find the probability that the computer contains neither a virus or a worm.

  4. Find the probability that the computer contains a virus but not a worm.

10.5.1.3. Practice Problem 2.1-C#

Reference: §2.1, Problem 14 in Navidi.

Six hundered paving stones were examined for cracks, and 15 were found to be cracked. The same 600 stones were then examined for discoloration, and 27 were found to be discolored. A total of 562 stones were neither cracked nor discolored. One of the 600 stones is selected from random.

  1. Find the probability that it is cracked, discolored, or both.

  2. Find the probability that it is both cracked and discolored.

  3. Find the probability that it is cracked but not discolored.

10.5.2. Counting Methods#

Further Reading: §2.2 in Navidi (2015)

10.5.2.1. Practice Problem 2.2-A#

Reference: §2.2, Problem 2 in Navidi.

A metallurgist is designing an experiment to determine the effect of flux, base metal, and energy input on the hardness of a weld. She wants to study four different fluxes, two different base metals, and three different amounts of energy input. If each run of the experiment involves one flux, one base metal, and one amount of energy input, how many runs are possible?

10.5.2.2. Practice Problem 2.2-B#

Reference: §2.2, Problem 4 in Navidi.

A group of 18 people have gotten together to play baseball. They will divide into two teams of 9 players each, with one team wearing green uniforms and the other wearning yellow uniforms. In how many ways can the people be divided into team?

# Add your solution here

Extension: How would your analysis change if there where 19 people, and we want to assign 9 people to the green team, 9 people to the yellow team, and one person to be the umpire?

10.5.2.3. Practice Problem 2.2-C#

Reference: §2.2, Problem 8 in Navidi.

In a certain state, license plates consist of three letters followed by three numbers.

  1. How many different license plates can be made?

  2. How many different license plates can be made in which no letter or number appears more than once?

  3. A license plate is chose at random. What is the probability that no letter or number appears more than once?

# Add your solution here

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10.5.2.4. Practice Problem 2.2-D#

Reference: §2.2, Problem 10 in Navidi.

A company has hired 15 new employees, and must assign 6 to the day shift, 5 to the graveyard shift, and 4 to the night shift. In how many ways can the assignment be made?

# Add your solution here

10.5.2.5. Practice Problem 2.2-E#

Reference: §2.2, Problem 12 in Navidi.

A drawer constains 6 red socks, 4 green socks, and 2 black socks. Two socks are chosen at random. What is the probability that they match?

10.5.3. Conditional Probability and Independence#

Further Reading: §2.3 in Navidi (2015)

10.5.3.1. Practice Problem 2.3-A#

Reference: §2.3, Problem 2 in Navidi

Let \(A\) and \(B\) be events with \(P(A) = 0.5\) and \(P(A \cap B^c) = 0.4\). For what values of \(P(B)\) will \(A\) and \(B\) be independent?

10.5.3.2. Practice Problem 2.3-B#

Reference: §2.3, Problem 6 in Navidi

The article “Integrating Risk…” estimates that 5.6% of a certain population has asthma, and that an asthmatic has probability 0.027 of suffering an asthma attack on a given day. A person is chosen at random from this populaton. What is the probability that this person has an asthma attack on that day?

10.5.3.3. Practice Problem 2.3-C#

Reference: §2.3, Problem 8 in Navidi

A drag racer has two parachutes, a main and a backup, that are designed to bring the vehicle to a stop after the end of a run. Suppose that the main chute deploys with probability 0.99, and that if the main fails to deploy, the backup deploys with probability 0.98.

  1. What is the probability that one of the two parachutes deploys?

  2. What is the probability that the backup parachute deploys?

10.5.3.4. Practice Problem 2.3-D#

Reference: §2.3, Problem 12 in Navidi

Sarah and Thomas are going bowling. The probability that Sarach scores more than 175 is 0.4, and the probability that Thomas scores more than 175 is 0.2. Their scores are independent.

  1. Find the probability that both score more than 175.

  2. Given that Thomas scores more than 175, the probability that Sarah scores more than Thomas is 0.3. Find the probability that Thomas scores more than 175 and Sarah scores more than Thomas.

10.5.3.5. Practice Problem 2.3-E#

Reference: §2.3, Problem 14 in Navidi

Laura and Philip each fire one shot at a target. Laura has probability 0.5 of hitting the target, and Philip has probability of 0.3. The shots are independent.

  1. Find the probability that the target is hit.

  2. Find the probability that the target is hit by exactly one shot.

  3. Given that the target was hit by exactly one shot, find the probability that Laura hit the target.

10.5.3.6. Practice Problem 2.3-F#

Reference: §2.3, Problem 26 in Navidi

A certain delivery service offers both express and standard delivery. Seventy-five percent of parcels are sent by standard delivery, and 25% are sent by express. Of those sent standard, 80% arrive the next day, and of those sent express, 95% arrive the next day. A record of parcel delivery is chosen at random from the company’s files.

  1. What is the probability that the parcel was shipped express and arrived the next day?

  2. What is the probability that it arrived the next day?

  3. Given the package arrived the next day, what is the probability that is was sent express?

10.5.4. Random Variables#

Further Reading: §2.4 in Navidi (2015)

10.5.4.1. Practice Problem 2.4-A#

Reference: §2.4, Problem 10 in Navidi.

Microprocessing chips are randomly sampled one by one from a large population, and tested to determine if they are acceptable for a certain application. Ninety percent of the chips in the population are acceptable.

  1. What is the probability that the first chip chosen is acceptable?

  2. What is the probability that the first chip is unacceptable, and the second is acceptable?

  3. Let \(X\) represent the number of chips that are tested up to and including the first acceptable chip. Find \(P(X=3)\).

  4. Find the probability mass function of \(X\).

10.5.4.2. Practice Problem 2.4-B#

Reference: §2.4, Problem 20 in Navidi.

The main bearing clearance (in mm) in a certain type of engine is a random variable with probability density function:

\[\begin{split} f(x)= \left\{ \begin{array}{ll} 625x & 0 < x \leq 0.04 \\ 50 - 625x & 0.04 < x \leq 0.08 \\ 0 & \mathrm{otherwise} \end{array} \right. \end{split}\]

  1. What is the probability that the clearance is less than 0.02 mm?

  2. Find the mean clearance.

  3. Find the standard deviation of the clearances.

  4. Find the cumulative distribution function of the clearance.

  5. Find the median clearance.

  6. The specification for the clearance is 0.015 to 0.063 mm. What is the probability that the specification is met?

10.5.5. Jointly Distributed Random Variables#

Reference: §2.6 in Navidi (2015)

10.5.5.1. Practice Problem 2.6-A#

Reference: §2.6, Problem 8 in Navidi.

The number of customers in line at a supermarket express checkout counter is a random variable whose probability mass function is given in the following table:

x

0

1

2

3

4

5

p(x)

0.10

0.25

0.30

0.20

0.10

0.05

For each customer, the number of items to be purchased is a random variable with probability mass function:

y

1

2

3

4

5

6

p(y)

0.05

0.15

0.25

0.30

0.15

0.10

Let \(X\) denote the number of customers in line, and let \(Y\) denote the total number of items purchased by all the customers in line. Assume the number of items purchased by one customer is independent of the number of items purchased by any other customer.

  1. Find \(P(X=2\) and \(Y=2)\).

  2. Find \(P(X=2\) and \(Y=6)\).

  3. Find \(P(Y=2)\).

10.5.5.2. Practice Problem 2.6-B#

Reference: §2.6, Problem 18 in Navidi.

A production facility contains two machines that are used to rework items that are initially defective. Let \(X\) be the number of hours that the first machine is in use, and let \(Y\) be the number of hours that the second machine is in use, on a randomly chosen day. Assume that \(X\) and \(Y\) have joint probability density function given by:

\[\begin{split} f(x)= \left\{ \begin{array}{ll} (3/2) (x^2 + y^2) & 0 < x <1 \ \mathrm{and} \ 0 < y < 1 \\ 0 & \mathrm{otherwise} \end{array} \right. \end{split}\]

  1. What is the probability that both machines are in operation for more than half an hour?

  2. Find the marginal probability density functions \(f_x(x)\) and \(f_y(y)\).

  3. Are \(X\) and \(Y\) independent? Explain.