STAT 120A/281A Homework 2

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STAT 120A/281A

Homework 2

1. For each of the following experiments, state the sample space S.

    a. A pair of dice is to be rolled, and the difference of the two numbers shown is to be recorded.

    b. Ask a randomly selected student how many days a week they commute to campus (assume students commute at least once a week).

    c. A coin is to be flipped 3 times, and each outcome is to be recorded (such as HHH).

    d. A coin is to be flipped 3 times, and the number of tails shown is to be recorded.

    e. A pair of dice is to be rolled, and the maximum of the two numbers shown is to be recorded.

2. a. Suppose P(A) = 0.48 and P(B) = 0.62.

Can A and B be mutually exclusive (yes, no, or maybe)? Why or why not.

b. Suppose P(A) = 0.48 and P(B) = 0.42.

Can A and B be mutually exclusive (yes, no, or maybe)? Why or why not.

3. Five applicants (Jim, Don, Mary, Sue, and Nancy) are available for two identical jobs. (Jim and Don are male; Mary, Sue, and Nancy are female.) A supervisor is to select two applicants to fill these jobs.

Abbreviate the names by J, D, M, S, and N.

    a. Write out the sample space associated with this experiment.

    b. Let A denote the event that exactly one female is selected. List the event/set A.

    c. Let B denote the event that at least one female is selected. List the event/set B.

    d. Write the event that two females are selected in terms of A and B.

    e. List the outcomes/elements in the event Ac.

4. On a large university campus, 84% of students report having a Microsoft product (phone, computer, mp3 player, etc.) and 33% report having an Apple product. Also, 86% report having a Microsoft or Apple product.

A student is randomly selected from this campus. Define the following events:

A = the randomly selected student has a Microsoft product

B = the randomly selected student has an Apple product.

    a. Write out the event that the randomly selected student has a Microsoft product and an Apple product. What is the probability of this event?

    b. Write out the event that the randomly selected student does not have a Microsoft product and does not have an Apple product. What is the probability of this event?

    c. Write out the event that the randomly selected student has either a Microsoft product or an Apple product, but not both. What is the probability of this event?

    d. What is the probability a randomly selected students has an Apple product but not a Microsoft product?

5. The following is called the Bonferroni’s inequality:

For events A and B, we have that

P(A ∩ B) ≥ P(A) + P(B) − 1.

    a. Prove the Bonferroni inequality.

    b. Let A and B be events with probabilities  and Show that 

6. A pair of dice are rolled once (so will have two numbers show up when we roll the pair). The sample space is S = {(1, 1),(1, 2),(1, 3), ....,(6, 5),(6, 6)}. There are 36 elements in the sample space, and they are all equally likely to occur (thus each has probability of  of occurring).

    a. What is the probability that the sum of the two dices is less than or equal to 11 ?

    b. What is the probability that both dices are even ?

7. This is not to be turned in, but you should do it. Run the following R code that will simulate the situation in 6a by rolling a pair of dice 10,000 times. Just copy and paste it into the R console, and press the enter key.

rolls = 10000

stor= c()

for (i in 1:rolls){

dice1 = sample(6,1)

dice2 = sample(6,1)

sum = dice1+dice2

stor[i] = ifelse(sum <= 11, 1, 0)

}

prob = sum(stor)/rolls

prob

Is this simulation value close to what you derived in question 6 part a?