Question 1

A fair red die and a fair blue die are rolled. What is the expected value of the sum of the number on the red die plus three times the number on the blue die?

Accepted Answer

Let $X _ { r }$ and $X _ { b }$ be the random variables for the numbers shown on the dice. We are asked for $E \left( X _ { r } + 3 X _ { b } \right)$. Since these dice are fair, we know that $E \left( X _ { r } \right) = E \left( X _ { b } \right) = ( 1 + 2 + 3 + 4 + 5 + 6 ) / 6 = 3.5$. By linearity of expectation, we have $E \left( X _ { r } + 3 X _ { b } \right) = 3.5 + 3 \cdot 3.5 = 14$.

Question 2

What is the probability that a fair coin lands heads four times out of ﬁve ﬂips?

Accepted Answer

There are 25 possible outcomes of ﬁve ﬂips. The number of possible ways to have the coin come up heads four times is the number of ways to pick four ﬂips out of ﬁve. This can be done in C(5, 4) = 5 ways. Hence the probability that the coin lands heads four times out of ﬁve ﬂips is 5/25 = 5/32.

Question 3

Two identical urns contain balls. One of the urns has 6 red balls and 3 blue balls. The other urn has 5

Accepted Answer

(a) Half the time we select the first urn, in which case the probability that the two balls are both red is $( 6 / 9 ) ( 5 / 8 ) = 5 / 12$. Half the time we select the second urn, in which case the probability that the two balls are both red is $( 5 / 13 ) ( 4 / 12 ) = 5 / 39$. Therefore the answer is
$$\frac { 1 } { 2 } \cdot \frac { 5 } { 12 } + \frac { 1 } { 2 } \cdot \frac { 5 } { 39 } = \frac { 85 } { 312 }$$ (b) Let $F$ be the event that the first ball is red, and let $S$ be the event that the second ball is red. We are asked for $p ( S \mid F )$. By definition, this is $p ( S \cap F ) / p ( F )$. In part (a) we found that $p ( S \cap F ) = 85 / 312$. By a simpler calculation, we see that
$$p ( F ) = \frac { 1 } { 2 } \cdot \frac { 6 } { 9 } + \frac { 1 } { 2 } \cdot \frac { 5 } { 8 } = \frac { 31 } { 48 }$$ Thus the answer is
$$\frac { 85 / 312 } { 31 / 48 } = \frac { 170 } { 403 }$$

Question 4

What is the probability that in a group of 200 random people, at least two of them have the same triple of initials (such as RSZ for Ruth Suzanne Zeitman), assuming that each triple of initials is equally likely. Give the answer as a calculation; it is not necessary to evaluate the expression.

Accepted Answer

The answer of What is the probability that in a...

Question 5

Suppose that a bag contains six slips of paper: one with the number 1 written on it, two with the number 2, and three with the number 3. What is the expected value and variance of the number drawn if one slip is selected at random from the bag?

Accepted Answer

The answer of Suppose that a bag contains six slips...

Question 6

What is the probability that a random person who tests positive for a certain blood disease actually has the disease, if we know that 1% of the population has the disease, that 95% of those who have the disease test positive for it, and 2% of those who do not have the disease test positive for it.

Accepted Answer

The answer of What is the probability that a random...

Question 7

What is the probability that a positive integer less than 100 picked at random has all distinct digits?

Accepted Answer

The answer of What is the probability that a positive...

Question 8

Two identical urns contain balls. One of the urns has 6 red balls and 3 blue balls. The other urn has 5 red

Accepted Answer

The answer of Two identical urns contain balls. One of...

Question 9

Suppose that two cards are drawn without replacement from a well-shuﬄed deck. What is the probability that both cards have numbers and that the numbers on the cards are the same (note that only the numbers 2 through 10 are shown on cards, since aces, kings, queens, and jacks are represented by letters).

Accepted Answer

The answer of Suppose that two cards are drawn without...

Quiz 7: Discrete Probability