Deck 1: Axioms of Peobability

Question 1

The waiting time (in seconds) between one song's end and the next song's beginning on a certain radio station is a random number between 3 and 6.8. Find the probability that the time between one song and another is at least 4.9 seconds.

Answer 1

$\frac { 1.8 } { 3.8 } = .47$

Question 2

Two fair 20-sided dice are rolled. How many total possible outcomes are there? What is the probability that the second number rolled is exactly 4 more than the first?

Answer 2

There are 400 total possible outcomes. The probability the second is 4 more than the first is 16/400.

Question 3

At the Internal Revenue Service, a social security number (9-digits) is selected randomly to receive an audit.
(a) What is the event that the last two digits are odd?
(b) What is the event that the last two digits form a number divisible by 5?

Answer 3

(a) $E = \left\{ \sum _ { i = 0 } ^ { 8 } a _ { i } \cdot 10 ^ { i } \mid a _ { i } \in \{ 0,1,2 , \ldots , 9 \} \text { for } i = 2 , \ldots , 8 , a _ { j } \in \{ 1,3,5,7,9 \} \text { for } j = 0,1 \right\}$
(b) $E = \left\{ \sum _ { i = 0 } ^ { 8 } a _ { i } \cdot 10 ^ { i } \mid a _ { i } \in \{ 0,1,2 , \ldots , 9 \} \text { for } i = 1 , \ldots , 8 , a _ { j } \in \{ 0,5 \} \text { for } j = 0 \right\}$

Question 4

Let Ω={a,b,c,d}. Choose two distinct 2-element subsets at random from P(Ω), the set of all subsets of . Find the size of the sample space of this experiment. Describe the following events explicitly:Sample Space: Let denote the sample space, then $| S | = 30$ .
(a) The two sets are disjoint.
(b) The intersection of the sets is {a}.
(c) The complement of the union of the sets consists of only the element d.

Question 5

At a given company there are 4 employees: Jim, Pam Dwight, and Michael. There is a breakroom and each employee can be inside or outside of the breakroom at any given time. Let $A _ { f } , A _ { p } , A _ { 0 } , \text { and } A _ { M }$ denote the event that Jim, Pam, Dwight, and Michael are in the breakroom, respectively. In terms of the events $A _ { f } , A _ { p } , A _ { D }$ , and $A _ { M } ,$ describe the event that at most two people are in the breakroom at a given time. What about the event that at least three people are in the breakroom?

Question 6

George likes antiques. He purchases an antique chair for $200 and plans to refurbish it. Give an explicit sample space for the resale value of his chair after he refurbishes it. Define, in set notation, the event that he loses money on his chair. Suppose George will only charge whole dollar amounts for convenience.

Question 7

Suppose we have a sample space Ω and two events A and B. Suppose A ⊂ B . Which statement is false? Show a counterexample for your answer.
(a) B cannot occur unless has occurred.
(b) A cannot occur unless has occurred.

Question 8

At a certain ice cream parlor, 40% of patrons get hot fudge, 25% do not get whipped cream, and 30% of patrons get both hot fudge and whipped cream. How many get hot fudge or whipped cream.

Question 9

A fair (6-sided) die is rolled until the first time two even numbers are rolled in a row. Give 10 elements of the sample space.

Question 10

A lion, who hunts only gazelle, buffalo, and giraffe, wakes up and hunts her first meal. If she is twice as likely to hunt a gazelle as a buffalo, and three times as likely to hunt a buffalo as a giraffe, find the respective probabilities of the lion hunting a gazelle, buffalo, and giraffe.

Question 11

In order to be crowned "Best in Town," John, a tennis player from Newton, MA must beat both Melissa and Jeffrey at tennis. The probability that he beats Melissa is 50% and the probability he beats Jeffrey is 61%. If the probability that he beats at least one of them is 93%, find the probability that John is crowned "Best in Town."

Question 12

Let $S = \left\{ \omega _ { 1 } , \omega _ { 2 } , \ldots \right\}$ be the sample space of some experiment. Let $m : S \rightarrow \mathbb { R }$ be given by $\operatorname { m } \left( \omega _ { n } \right) = \frac { 1 } { 3 ^ { n } }$ . Can we define a probability distribution on P so S that $P \left( \left\{ \omega _ { n } \right\} \right) = m \left( \omega _ { n } \right)$

Question 13

Jeff has 3 pens, which he uses to do professional illustrations. He has a red pen, a blue pen, and a green pen. He often works on illustrations outside of home, and it is important that all pens are at least half-full with ink. Suppose a pen can be full, half-full or empty and that the level of ink in a pen is independent of the level of ink in other pens. How large is the sample space of this experiment? Describe the event that all three pens are at least half-full.

Question 14

For two events and explain why the following are impossible:i. $P ( A B ) = 2 P ( A )$ ii. $P ( A \cup B ) = P ( A ) - P ( B )$

Question 15

A certain auto shop receives 7 autos, each on a random day, in the next seven days. The Smiths drop off all three of their vehicles while the Jones and the Masoods each drop off two vehicles. Write a sample space for the number of days between the day the Smiths drop off a second vehicle and the day they drop off a third vehicle (for instance if they drop off their second vehicle on the third day and their third on the fourth day, one day is between these drop off times.) From this sample space, describe the event that the Jones drop off their two vehicles on the last two days of this period.