7.3 Review Exercices

1. A company has created a table to evaluate how long new hires remain with the company. Complete the following table using the data provided.

[latex]x[/latex]	[latex]P(x)[/latex]
[latex]0[/latex]	[latex]0.12[/latex]
[latex]1[/latex]	[latex]0.18[/latex]
[latex]2[/latex]	[latex]0.30[/latex]
[latex]3[/latex]	[latex]0.15[/latex]
[latex]4[/latex]
[latex]5[/latex]	[latex]0.10[/latex]
[latex]6[/latex]	[latex]0.05[/latex]

a) [latex]P(x=4)[/latex]

b) [latex]P(x\geq4)[/latex]

c) On average, how long would you expect a new hire to stay with the company?

d) What does the column [latex]P(x)[/latex] sum to?

Solution

a) [latex]0.10[/latex]

b) [latex]0[/latex]

c) Two years

d) [latex]1[/latex]

 

 

2. A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell to everyone and no fewer. Through observation, the baker has established a probability distribution.

[latex]x[/latex]	[latex]P(x)[/latex]
[latex]1[/latex]	[latex]0.15[/latex]
[latex]2[/latex]	[latex]0.35[/latex]
[latex]3[/latex]	[latex]0.40[/latex]
[latex]4[/latex]	[latex]0.10[/latex]

a) Define the random variable [latex]x[/latex].
b) What is the probability the baker will sell more than one batch?
c) What is the probability the baker will sell exactly one batch?
d) On average, how many batches should the baker make?

Solution

a) The number of batches the baker makes.

b) [latex]0.35 + 0.40 + 0.10 = 0.85[/latex]

c) [latex]15%[latex] d) [latex]1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45[/latex]

 

 

3. Ellen has music practice three days a week. She practices for all of the three days [latex]85%[/latex] of the time, two days [latex]8%[/latex] of the time, one day [latex]4%[/latex] of the time, and no days [latex]3%[/latex] of the time. One week is selected at random.

a) Define the random variable [latex]x[/latex].

b) Construct a probability distribution table for the data.

Solution

a) The amount of the three days which Ellen practices.,

b)

[latex]x[/latex]	[latex]P(x)[/latex]
[latex]0[/latex]	[latex]0.03[/latex]
[latex]1[/latex]	[latex]0.04[/latex]
[latex]2[/latex]	[latex]0.08[/latex]
[latex]3[/latex]	[latex]0.85[/latex]

 

 

4. We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic?

Solution

Each probability is between zero and one.

 

 

5. Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events [latex]35%[/latex] of the time, four events [latex]25%[/latex] of the time, three events [latex]20%[/latex] of the time, two events [latex]10%[/latex] of the time, one event [latex]5%[/latex] of the time, and no events [latex]5%[/latex] of the time.

a) Define the random variable [latex]x[/latex].

b) What values does [latex]x[/latex] take on?

c) Construct a probability distribution table with the data.

d) Find the probability that Javier volunteers for less then three events each month.

e) Find the probability that Javier volunteers for at least one event each month.

Solution

a) The number of events Javier volunteers for each month

b) Discrete values between [latex]0[/latex] and [latex]5[/latex],

c)

[latex]x[/latex]	[latex]P(x)[/latex]
[latex]0[/latex]	[latex]0.05[/latex]
[latex]1[/latex]	[latex]0.05[/latex]
[latex]2[/latex]	[latex]0.10[/latex]
[latex]3[/latex]	[latex]0.25[/latex]
[latex]4[/latex]	[latex]0.35[/latex]

d) [latex]0.05+0.05+0.10=0.20[/latex]

e) [latex]1-0.05=0.95[/latex]

 

 

6. Suppose that the probability distribution function for the number of years it takes to earn a Bachelor of Science (B.S.) degree in given in the following table.

[latex]x[/latex]	[latex]P(x)[/latex]
[latex]3[/latex]	[latex]0.05[/latex]
[latex]4[/latex]	[latex]0.40[/latex]
[latex]5[/latex]	[latex]0.30[/latex]
[latex]6[/latex]	[latex]0.15[/latex]
[latex]7[/latex]	[latex]0.10[/latex]

a) Define the random variable [latex]x[/latex].

b) What does it mean that the values zero, one, and two are not included for [latex]x[/latex] in the PDF?

Solution

a) The number of years it takes to earn a B.S. degree

b) Nobody sampled has acquired a B.S. in [latex]0,1,\text{ or }2[/latex] years.

 

 

7. Complete the expected value table.

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]
[latex]0[/latex]	[latex]0.2[/latex]
[latex]1[/latex]	[latex]0.2[/latex]
[latex]2[/latex]	[latex]0.4[/latex]
[latex]3[/latex]	[latex]0.2[/latex]

Solution

[latex]x*P(x)[/latex]

[latex]0(0.2)=0[/latex]

[latex]1(0.2)=0.2[/latex]

[latex]2(0.4)=0.8[/latex]

[latex]3(0.2)=0.6[/latex]

 

 

8. Find the expected value from the expected value table.

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]
[latex]2[/latex]	[latex]0.1[/latex]	[latex]0.2[/latex]
[latex]4[/latex]	[latex]0.3[/latex]	[latex]1.2[/latex]
[latex]6[/latex]	[latex]0.4[/latex]	[latex]2.4[/latex]
[latex]8[/latex]	[latex]0.2[/latex]	[latex]1.6[/latex]

Solution

[latex]0.2 + 1.2 + 2.4 + 1.6 = 5.4[/latex]

 

 

9. Find the standard deviation

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]	[latex](x-\mu)^2*P(x)[/latex]
[latex]2[/latex]	[latex]0.1[/latex]	[latex]2(0.1)=0.2[/latex]	[latex](2-5.4)^2*0.1=1.156[/latex]
[latex]4[/latex]	[latex]0.3[/latex]	[latex]4(0.3)=1.2[/latex]	[latex](4-5.4)^2*0.3=0.588[/latex]
[latex]6[/latex]	[latex]0.4[/latex]	[latex]6(0.4)=2.4[/latex]	[latex](6-5.4)^2*0.4=0.144[/latex]
[latex]8[/latex]	[latex]0.2[/latex]	[latex]8(0.2)=1.6[/latex]	[latex](8-5.4)^2*0.2=1.352[/latex]

Solution

[latex]\frac{1.156+0.588+0.144+1.352}{4}=0.81[/latex]

 

 

10. Identify the mistake in the probability distribution table

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]
[latex]1[/latex]	[latex]0.15[/latex]	[latex]0.15[/latex]
[latex]2[/latex]	[latex]0.25[/latex]	[latex]0.50[/latex]
[latex]3[/latex]	[latex]0.30[/latex]	[latex]0.90[/latex]
[latex]4[/latex]	[latex]0.20[/latex]	[latex]0.80[/latex]
[latex]5[/latex]	[latex]0.15[/latex]	[latex]0.75[/latex]

Solution

The values of [latex]P(x)[/latex] do not sum to [latex]1[/latex]

 

 

11. A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. He has the following probability distribution.

[latex]x[/latex]	[latex]P(x)[/latex]
[latex]1[/latex]	[latex]0.35[/latex]
[latex]2[/latex]	[latex]0.20[/latex]
[latex]3[/latex]	[latex]0.15[/latex]
[latex]4[/latex]
[latex]5[/latex]	[latex]0.10[/latex]
[latex]6[/latex]	[latex]0.05[/latex]

a) Define the random variable [latex]x[/latex].

b) Define [latex]P(x)[/latex], or the probability of [latex]x[/latex].

c) Find the probability that a physics major will do post-graduate research for four years.

d) Find the probability that a physics major will do post-graduate research for at most three years.

e) On average, how many years would you expect a physics major to spend doing post graduate research?

Solution

a) The amount of years a student will spend doing post-graduate research.

b) The probability that a student will spend [latex]x[/latex] amount of years doing post-graduate research

c) [latex]1-0.35-0.20-.15-0.05=0.15[/latex]

d) [latex]1-0.35-0.20-0.15-0.10-0.05=0.15[/latex]

e) [latex]1(0.35)+2(0.20)+3(0.15)+4(0.15)+5(0.10)+6(0.05)=0.35+0.40+0.45+0.60+0.50+0.30=2.6 years[/latex]

 

 

12. A venture capitalist, willing to invest [latex]$1,000,000[/latex], has three investments to choose from. The first investment, a software company, has a [latex]10%[/latex] chance of returning [latex]$5,000,000[/latex] profit, a [latex]30%[/latex] chance of returning [latex]$1,000,000[/latex] profit, and a [latex]60%[/latex] chance of losing the million dollars. The second company, a hardware company, has a [latex]20%[/latex] chance of returning [latex]$3,000,000[/latex] profit, a [latex]40%[/latex] chance of returning [latex]$1,000,000[/latex] profit, and a [latex]40%[/latex] chance of losing the million dollars. The third company, a biotech firm, has a [latex]10%[/latex] chance of returning [latex]$6,000,000[/latex] profit, a [latex]70%[/latex] of no profit or loss, and a [latex]20%[/latex] chance of losing the million dollars.

a) Construct a probability distribution function for each investment.

b) Find the expected value for each investment.

c) Which is the safest investment? Why do you think so?

d) Which is the riskiest investment? Why do you think so?

e) Which investment has the highest expected return, on average?

Solution

a)

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]
[latex]5,000,000[/latex]	[latex]0.10[/latex]	[latex]500,000[/latex]
[latex]1,000,000[/latex]	[latex]0.30[/latex]	[latex]300,000[/latex]
[latex]0[/latex]	[latex]0.60[/latex]	[latex0][/latex]

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]
[latex]3,000,000[/latex]	[latex]0.20[/latex]	[latex]600,000[/latex]
[latex]1,000,000[/latex]	[latex]0.40[/latex]	[latex]400,000[/latex]
[latex]0[/latex]	[latex]0.40[/latex]	[latex]0[/latex]

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]
[latex]6,000,000[/latex]	[latex]0.10[/latex]	[latex]600,000[/latex]
[latex]1,000,000[/latex]	[latex]0.70[/latex]	[latex]700,000[/latex]
[latex]0[/latex]	[latex]0.20[/latex]	[latex]0[/latex]

b) Expected value of [latex]800,000[/latex] for the software company, [latex]1,000,000[/latex] for the hardware company, and [latex]1,300,000[/latex] for the biotech firm
c) The safest investment would be in the biotech firm. Justifications include there being the lowest probability for losing the investment, and the highest potential for profit.
d) The riskiest investment would be in the software company. Justifications include there being the highest possibility for losing the investment, and the lowest potential for profit.
e) On average, the biotech firm has the highest expected return.

 

13. Suppose that [latex]20,000[/latex] married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let [latex]x[/latex] be the number of children married people have.

[latex]x[/latex]	[latex]P(x)[/latex]	[latex]x*P(x)[/latex]
[latex]0[/latex]	[latex]0.10[/latex]
[latex]1[/latex]	[latex]0.20[/latex]
[latex]2[/latex]	[latex]0.30[/latex]
[latex]3[/latex]
[latex]4[/latex]	[latex]0.10[/latex]
[latex]5[/latex]	[latex]0.05[/latex]
[latex]6\text{(or more)}[/latex]	[latex]0.05[/latex]

a) Find the probability that a married adult has three children.

b) In words, what does the expected value in this example represent?

c) Find the expected value.

d) Is it more likely that a married adult will have two to three children or four to six children? How do you know?

Solution

a) [latex]1-0.05-0.05-0.10-0.30-0.20-0.10=0.20[/latex]

b) The average amount of children that the survey participants had.

c) [latex]0(0.10)+1(0.20)+2(0.30)+3(0.20)+4(0.20)+5(0.05)+6(0.05)=2.25[/latex]

d) The probability of a married adult having two to three children is [latex]0.30+0.20[/latex] according to this table. The probability of a married adult having four to six children is [latex]0.10+0.05+0.05=0.20[/latex] according to this table. Therefore, it is more probable that a married adult will have two to three children.

 

 

14. Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given as in following table. On average, how many years do you expect it to take for an individual to earn a B.S.?

[latex]x[/latex]	[latex]P(x)[/latex]
[latex]3[/latex]	[latex]0.05[/latex]
[latex]4[/latex]	[latex]0.40[/latex]
[latex]5[/latex]	[latex]0.30[/latex]
[latex]6[/latex]	[latex]0.15[/latex]
[latex]7[/latex]	[latex]0.10[/latex]

Solution

[latex]3(0.05)+4(0.40)+5(0.30)+6(0.15)+7(0.10)=4.85[/latex]