Math 2: Algebra 2, Geometry and Statistics ...lps.lexingtonma.org/cms/lib2/MA01001631/Centricity/Domain/835... · probability!of!drawing!anevennumbered!card,!thendrawing!a!10?!!!!!

Math 2: Algebra 2, Geometry and Statistics

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

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Chapter 5 Test Review – Probability – Math 2 Students Will Be Able To:

• Calculate probabilities of random events. Rolling a number cube, flipping a coin, spinning a spinner etc.

• Calculate probabilities of conditional events (spinner, dice, etc. or from data table). P(A|B) • Calculate probabilities of compound events (both independent and dependent cases) P(A and B) • Write out a sample space and determine the total number of outcomes for given situation • Determine if two events are independent à is it a situation that you can explain, i.e. there is no

replacement so the first event necessarily affects the total left so they must be dependent, or à you need to calculate and show that P(A) = P(A|B) or P(B) = P(B|A). This is needed when you are working with data tables.

• Complete a data table given that two events are independent. • Find the total number of arrangements of things (both binary like flipping a coin and non-‐binary like

people sitting at a table or letters in a word) • Generate and use Pascal’s triangle to assist you solving problems. • Find the expected value for a given situation. • Determine if outcomes are equally likely. • Use the binomial theorem to expand (x + y)n

Practice Problems 1.

a. If you flip a coin five times, how many different ways are there for the result to be 3 heads and 2 tails? Write them out.

b. Calculate the value of !! . Explain how it relates to the work in part (a).

2. Consider flipping a coin eight times.

a. Find the probability you will get exactly five heads and three tails. b. Find the probability you will get exactly four heads and four tails.



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3. In a game, you roll a standard number cube and flip a coin. The coin has the number 2 on one side, and 6 on the other side. Your score is the sum of the values that appear on the number cube and the coin-‐flip.

a. Use a table to write out the entire sample space for this experiment. b. You win the game if you score 8 points or more. Find the probability that you win. c. You get to roll the number cube and flip the coin a second time if you score 5. Find the probability that you score 5. d. Are all the scores from 3 to 12 equally likely? Explain.

4.

a. When drawing two cards from a standard deck of card without replacement, what is the probability of drawing an even numbered card, then drawing a 10?

b. When drawing two cards from a standard deck of card with replacement, what is the probability of drawing an even numbered card, then drawing a 10?

5. An urn contains 7 blue marbles, 5 red marbles, and 2 white marbles. If you draw two marbles,

what is the probability that they are different colors? There are two possible scenarios here; calculate the probability in each case and explain how the cases are different.



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Note:

!𝑎𝑏! = !𝐶!

6. A student guesses randomly on a 6-‐question TRUE/FALSE test.

a. How many outcomes are possible for answering the test (e.g. FFTTTT)?

b. What is the chance the student gets 3 right and 3 wrong? 7. Finish writing out Pascal’s Triangle to the 8th row using both choose notation and values.

!! = 1

!! = 1

!! = 1

!! = 1

!! = 2

!! = 1

!! = 1

!! = 3

!! = 3

!! = 1



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8. A local gym wants to gather data on the levels of gym membership among adult men and women in the community. The table below summarizes the data that they found.

Women (W) Men (M) Total

Gym Members (G) 80

Non Gym Members (N)

Total 90 240

a. Complete the table so that the number of men with gym memberships is the same as the

number of women with gym memberships.

b. Find P(M)

c. Find P(G)

d. Find P(M|G)

e. Find P(N|W)

f. Are being a gym member and being a woman independent? Support your answer.

g. Complete the table so that being a man and being a gym member are independent.

Women (W) Men (M) Total

Gym Members (G) 80

Non Gym Members (N)

Total 90 240



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9. The freshman class of a local high school was given a survey in which there were two questions, “Do you play soccer?” and “Are you a fan of the Soccer World Cup.” The results are summarized in the table below.

Serious Fan (S) Casual Fan (C) Not a Fan (N) Total

Plays Soccer (P) 146 24 30 200

Doesn’t Play (D) 14 36 250 300

Total 160 60 280 500

a. Are being a serious fan (S) and playing soccer (P) independent? Show calculations that support your answer.

b. Are being a casual fan (C) and playing soccer (P) independent? Show calculations that support your answer.

c. Are not being a fan (N) and playing soccer (P) independent? Show calculations that support your answer.

d. Outcomes that are not independent are said to be Associated. Write a one-‐sentence summary that explains what variables are associated.



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10. Jonah has five different plants and five pots, each labeled with one of the plant names. He does not know the correct name of each plant so he randomly puts one plant in each pot. What is the probability that all five of the plants are in the pot with the correct name?

11. A snack machine is malfunctioning. Sometimes when you put the cost of a snack in the machine, it

gives you more than 1 snack, or it gives you no snack at all. The probabilities are shown in the table.

a. Calculate the expected value for how many snacks you get.

b. On average, is the machine giving out more snacks than it should, or less snacks than it should? Explain how you know.

12. Write the expansion of (𝑎 + 𝑏)!. Explain how this is related to Pascal’s triangle.

X = how many snacks you get Probability 0 0.04 1 0.92 2 0.03 3 0.01



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13. A state runs a lottery called “Pick 6.” • A player buys a $1 lottery ticket showing 6 of the numbers from 1 through 42. • Then a set of 6 numbers is chosen as the winning numbers, and anyone whose ticket matches

those numbers wins a $2,000,000 prize. • The table below lists the probability of making each number of matches.

Matches Frequency Payout

0 correct 1,947,792 $0

1 correct 2,261,952 $0

2 correct 883,757 $0

3 correct 142,800 $1

4 correct 9450 $75

5 correct 216 $1,500

6 correct 1 $2,000,000

Total

a. Find the expected value of one lottery ticket.

b. On average, how much money does the state earn on a lottery ticket. 14.

a. How many ways can the letters of HOUSE be arranged? Show your calculations.

b. How many ways can the letters of BOOBOO be arranged? Show your calculations.



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15. Let A and B represent any two events having non-‐zero probabilities.

a. Under what circumstances would P(B | A) = P(B)?

b. Under what circumstances would P(B | A) = 0? 16. A jar contains red marbles (®) and chartreuse (greenish) marbles (©) in the quantities shown

below. ® ® ® ® © © ©

Two marbles are randomly drawn from the jar, one after the other without replacement.

a. What is the probability that the first marble is chartreuse and the second marble is red?

b. Define the events A = “first marble is chartreuse” and B = “second marble is red.” Are these two events independent? Justify your answer.

17. Consider the number of loudspeaker announcements per day at school. Suppose there’s a 15%

chance of having 0 announcements, a 30% chance of having 1 announcement, a 25% chance of having 2 announcements, a 20% chance of having 3 announcements, and a 10% chance of having 4 announcements. Find the expected value of the number of announcements per day.



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ANSWERS

1. a. 10, HHHTT, HHTTH, HTTHH, TTHHH, HHTHT, HTHTH, THTHH, HTHHT, THHTH, THHHT.

b. !! =10. You have two tails, and you are choosing which of the 5 flips for them to happen.

2. a. 𝑃 =!!!!= !"

!"#

b. 𝑃 =!!!!= !"

!"#

3.

a. DICE ROLL

1 2 3 4 5 6 COIN FLIP

2 1+2 = 3 2+2=4 3+2=5 4+2=6 5+2=7 6+2=8 6 1+6=7 2+6=8 3+6=9 4+6=10 5+6=11 6+6=12

b. P(8 or more) = !

!"

c. P (5) = !!"

d. No, 7 and 8 are twice as likely as any of the other outcomes.

4. a. P(2,4, 6, or 8 then 10)=!"

!"∗ !!"= !"

!!", and P(10 then 10) = !

!"∗ !!"= !

!!"

so P(even then 10)= !"!!"

b. P(even then 10)=!"!"∗ !!"= !

!"#

5. The two cases are with replacement and without replacement.

a. With replacement

First Marble Blue First Marble Red Two White Marbles TOTAL

714

∗714

=49196

514

∗914

=45196

214

∗1214

=24196

49196

+45196

+24196

=118196

=5998

b. Without replacement

First Marble Blue First Marble Red Two White Marbles TOTAL

714

∗713

=49182

514

∗913

=45182

214

∗1213

=24182

49182

+45182

+24182

=118182

=5991



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6. a. 26=64

b. !!!!= !"

!"= !

!"

7.

!! = 1

!! = 4

!! = 6

!! = 4

!! = 1

!! = 1

!! = 5

!! = 10

!! = 10

!! = 5

!! = 1

!! = 1

!! = 6

!! = 15

!! = 20

!! = 15

!! = 6

!! = 1

!! = 1

!! = 7

!! = 21

!! = 35

!! = 35

!! = 21

!! = 7

!! = 1

!! = 1

!! = 8

!! = 28 8 = 56 !

! = 70 !! = 56

!! = 28

!! = 8

!! = 1

8.

a. Women (W) Men (M) Total

Gym Members (G) 40 40 80

Non Gym Members (N) 50 110 160

Total 90 150 240

b. P(M) = !"#!"#

c. P(G) = !"

!"#

d. P(M|G) = !"

!"

e. P(N|W) = !"

!"

f. No, in order to be independent, both P(W) = P(W|G) and P(G) = P(G|W). Both of these are

false. g.

Women (W) Men (M) Total Gym Members

(G) 30 50 80

Non Gym Members (N) 60 100 160

Total 90 150 240



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9. a. If S and P are independent, both of the following would be true:

• P(S) = P(S|P) and P(P) = P(P|S). • P(S) = 160/500 = 0.32; P(S|P) = 146/200 = 0.73 and • P(P) = 200/500 = 0.4 and P(P|S) = 146/160 = 0.9125 • Therefore S and P are NOT independent.

b. If C and P are independent, both of the following would be true: • P(C) = P (C|P) and P(P) = P(P|C) • P(C) = 60/500 = 0.12 and P(C|P) = 24/200 = 0.12 • P(P) = 200/500 = 0.4 and P(P|C) = 24/60 = 0.4 • Therefore C and P ARE independent.

c. If N and P are independent, both of the following would be true: • P(N) = P(N|P) and P(P) = P(P|N) • P(N) = 280/500 = 0.56 and P(N|P) = 30/200 = 0.15 • P(P) = 200/500 = 0.4 and P(P|N) = 30/280 = 0.1071 • Therefore N and P are NOT independent

d. Whether someone is a serious fan is associated with whether someone plays soccer. OR Whether someone is not a fan is associated with whether someone plays soccer.

10. There are 5! = 120 ways to arrange the plants and only one way where all the plants are in the correct pots. Therefore the probability is !

!"#.

11. a.

b. On average the machine is giving out more snacks than it should. It should give out exactly one snack each time, but it is averaging a little higher than that.

12. (𝑎 + 𝑏)! = 𝑎! + 6𝑎!𝑏 + 15𝑎!𝑏! + 20𝑎!𝑏! + 15𝑎!𝑏! + 6𝑎𝑏! + 𝑏!. The coefficients of each term in this expansion are the numbers in the 6th row of Pascal’s triangle. The exponents of each of the terms correspond to the combination numbers that generate Pascal’s triangle. For example, 15𝑎!𝑏! corresponds to !! =

!! = 15.

13. a. E ≈ $0.6054 b. The state makes about 40 cents on each lottery ticket

14. a. 5*4*3*2*1 = 5! = 120 b. !!

!!!!= !

! = !! = 15

15. a. If A and B are independent b. If A and B are mutually exclusive

X = how many snacks you get Probability Value 0 0.04 0 1 0.92 0.92 2 0.03 0.06 3 0.01 0.03 TOTAL 1.01



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16. a. !

!∗ !!= !"

!"

b. No, if the first marble is chartreuse, there is a better chance of the second marble being red than if the first marble is not chartreuse.

17. E = 1.8 announcements per day.

Documents

Math 2: Algebra 2, Geometry and Statistics ...lps.lexingtonma.org/cms/lib2/MA01001631/Centricity/Domain/835... · probability!of!drawing!anevennumbered!card,!thendrawing!a!10?!!!!!