Chapter 6, The mathematics of probability Video Solutions, An Introduction to Decision Theory

Problem 1

A card deck contains 52 cards, half of which are red and half of which are black. You randomly draw a card, put it back, and then draw a second card.
(a) What is the probability of drawing two red cards?
(b) What is the probability of drawing two red or two black cards?
(c) What is the probability of drawing one red and one black card?

James Kiss

Numerade Educator

View

Problem 2

You draw two cards from a card deck, without putting back the first before drawing the second.
(a) What is the probability of drawing two red cards?
(b) What is the probability of drawing two red or two black cards?
(c) What is the probability of drawing one red and one black card?
(d) What is the probability that the second card is red given that the first is red?
(e) What is the probability that the second card is red given that the first is black?

James Kiss

Numerade Educator

02:11

Problem 3

The probability that the next president will be a democrat is $1 / 2$, and the probability that the next president will be a woman is 0.3 , but the probability that the next president will be a woman and a democrat is just 0.1 . What is the probability that the next president will be democrat or a woman?

Gregory Higby

Numerade Educator

01:23

Problem 4

You know the following: $p(\neg A)$ is 0.6 , and $p(B)$ is 0.3 , and $p(A \rightarrow B)$ is
0.7 . What is $p(\neg A \wedge B)$ ?

Maxime Rossetti

Numerade Educator

00:20

Problem 5

You roll a fair die twice. What is the probability that the total sum will exceed 5 , given that the first roll is 1 ?

Ankit Gupta

Numerade Educator

01:29

Problem 6

Two fair six-sided dice are rolled. (a) What is the probability that their sum is 7 ? (b) Does the probability that the sum is 7 depend on the score
shown on the first die?

Manisha Sarker

Numerade Educator

01:23

Problem 7

You know the following: $p(A)=0.1$, and $p(B \mid A)=1$, and $p(\mathrm{~B} \neg \mathrm{A})=0.3$. What is $p(\mathrm{AlB})$ ?

Maxime Rossetti

Numerade Educator

View

Problem 8

According to an article in a computer magazine $5.7 \%$ of all computers break down during their predicted lifespan, and $9.5 \%$ of all computers are infected by a virus. Furthermore, $92 \%$ of all computers that break down are infected by a virus. What is the probability that a computer that is infected by a virus breaks down?

Keondre Parker

Numerade Educator

03:27

Problem 9

Your doctor suspects that you may be suffering from a rare disease that affects about 1 in 50,000. Given that you have the disease, the test offered by the doctor will show positive with probability 0.9 . Unfortunately, the test will also show positive with probability 0.01 when applied to a healthy person. What is the probability that you have the disease given that the test is positive? What is your opinion about the test offered by the doctor?

William Mead

Numerade Educator

01:00

Problem 10

You toss a coin three times, and it lands heads up every time. You're apriori probability that the coin is biased to land heads is 0.1 . What is your aposteriori probability that the coin is biased to always land heads up?

Crystal Wang

Numerade Educator

03:53

Problem 11

Your a-priori probability that the coin is biased is 0 . Prove that no matter how many times you toss it and update the probability in light of the information received, the probability can never become nonzero.

Manisha Sarker

Numerade Educator

01:37

Problem 12

Prove that if $A$ is independent of $A$, then $p(A)=0$ or $p(A)=1$.

Manisha Sarker

Numerade Educator

04:27

Problem 13

What is the probability that at least two children in a group of $n$ children, all born in 2009, share a birthday? (Why did I assume that all children were born in 2009?)

Muhammad Nawaz

Numerade Educator

01:47

Problem 14

Rumour has it that the following question was asked many years ago to students applying to study philosophy at the University of Cambridge: "Four witnesses, A, B , C and D, at a trial each speak the truth with probability $\frac{1}{3}$ independently of each other. In their testimonies, A claimed that $\mathrm{B}$ denied that $\mathrm{C}$ declared that $\mathrm{D}$ lied. What is the conditional probability that $\mathrm{D}$ told the truth?"

Mahipal Kumawat

Numerade Educator