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Name:__________________________

Probability Unit

# Assignment Completed? Comments1. Applications of Probability2. Assignment 1 – pg 5 - 73. Theoretical Probability4. Assignment 2 – pg 9 - 115. Experimental Probability6. Assignment 3 – pg 13-167. Compounding Independent

Events8. Assignment 4 – pg 17 - 199. Probability and Odds10

.Assignment 5 - pg 23, 24

11.

Expected Value

12.

Assignment 6 – pg 26 - 28

Probability Test:_________________________

Lesson #1: Applications of Probability

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Decisions are often based on the probability of something occurring. Doctors research the probability of a medicine curing an ailment before prescribing medicine. Insurance companies examine the probability of unnatural events before setting insurance rates. Probability can be written as a fraction or as a decimal.

Probability of an event = number of ways the event can occur total number of possible events

Let’s review fractions and ratios:

For the following fractions, reduce the fraction to its simplest form, write the fraction as a decimal, write the fraction as a percentage.

Reduce Decimal Percent

25 / 80

6 / 60

50 / 50

Consider the box beside, write each ratio. > [ > > [ [ [ > > >

Ratio > : [ =

[ : > =

> : all=

[ : all =

Probability is the chance that something will happen. Percentages are often associated with probability to illustrate how likely it is that an event will happen out of 100. The number of ways an event can occur (numerator) is always less than or equal to the total number of possible events (denominator). Therefore, probability is always a value between 0 and 1.

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The more likely an event is to occur, the closer its probability will be to 1.

The less likely an event is to occur, the closer its probability is to 0. A probability of 1 means an event is certain to occur, a probability of 0

means that even will never occur.

Ex. #1 A city is interested in knowing how many of its roads and sidewalks need to be repaired. A city engineer randomly inspects 38 city roads and finds 4 that need to be repaired. He also finds that 6 of the 45 sidewalks he inspects need to be repaired.

a. What is the probability a city road has to be repaired? Express this as a decimal.

b. What is the probability a city sidewalk has to be repaired? Express this as a decimal.

c. What do we know about city roads and sidewalks?

Ex. #2 A local airline offers its passengers a choice of two different types of meals on its dinner flights, vegetarian lasagna or meat lasagna. On its flights last year, approximately 22 250 out of 30 000 passengers chose the meat lasagna.

a. Based on this data, what is the probability of a passenger choosing meat lasagna on future flights? Express your answer as a decimal, percent, and a ratio.

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b. Based on this probability, how many vegetarian lasagna meals should the airline order for a dinner flight that has 228 passengers booked?

Ex. #3 The probability that it rains tomorrow is 40%. What is the probability that it does not rain tomorrow?

Assignment #1

1. Change the following decimals to percent.

.58 .02

.32 .985

2. Change the following percentages to decimal.

100% 2.45%

40% 66.5%

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3. Place the following events on a probability scale.

a. You will live for 100 years.

b. In the first week of November, it will snow at least once in Winnipeg.

c. The sun will rise in the east tomorrow morning.

d. The next baby born in Victoria Hospital will be a girl.

e. You will get 80% or higher in this unit.

Impossible l---------------------------------------------------------------------l Certain

0 1

4. If you were told that the probability of an airplane crashing on any flight is 0.0005%, would you consider this an accurate percentage? Why? Would you get on this plane? Why or Why not? Would you get on the plane if this percentage increased to 0.005%. Why or Why not?

5. The probability of winning a particular Lotto 6 / 49 is approximately 1 in 14 000 000. The probability of being hit by lightning once in your lifetime is approximately 1 in 600 000. Are you more likely to win the lottery or be hit by lightning?

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6. The probability that a person is left-handed is 1 in 10 people. The city of Winnipeg has a population of approximately 650 000 people. How many people are left-handed in Winnipeg?

7. The probability of twins being born is 1:90. If during one year there were 15 500 births in a particular city, how many of those births would be twins?

8. An entrepreneur determines that 2% of her business is lost due to bad debt. If she bills $68 500 during a year, approximately how much will she lose due to bad debt?

9. The weather stations indicates that there is a 30% chance of rain on Tuesday. What is the probability that it will not rain?

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Lesson #2: Theoretical Probability

The Theoretical Probability of an event occurring is based on theory. That is, what do we expect to happen in a perfect world. We can find out the theoretical probability of an event occurring by using the following ratio:

Probability of an event = number of ways the event can occur total number of possible events

Ex. #1 What is the probability of rolling each of the following with a die.

a. The number 5

b. An even number.

c. A number greater than 4.

d. Not the number 3.

e. The number 9.

Assignment #2:

1. What is the probability of rolling each of these numbers with a fair die?

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a. The number 3

b. An odd number

c. A number greater than or equal to 2.

d. Not the number 5 or 6.

e. The number 0.

2. The diagram below shows a jar of jelly beans. The jelly beans are of the same size and shape. You put your hand in to the jar and without looking select a jelly bean. What is the probability of each of the following? Express each probability as a fraction and as a percent!

a. P (yellow)

b. P (Green)

c. P (Yellow or Green)

d. P (not black)

e. P (not white)

3. Each letter of the word MATHEMATICAL is on a different card. All the cards are the same size. The cards are placed face down and shuffled. What is the probability that you will draw each of the following?

a. P (M)

b. P (A)

c. P (C, H, or L)

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d. P (not T)

e. P (vowel)

4. What is the probability of spinning each of the following with the given spinner? Express each probability as a fraction and as a percent.

a. P (red)

b. P (yellow)

c. P (red, yellow, or green)

d. P (neither red nor yellow)

e. P (not blue)

5. What is the probability of drawing the following cards from a standard deck of 52 playing cards?

a. P (5 of clubs)

b. P (diamond)

c. P (red card)

d. P (ace)

e. P (a black 10)

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6. There are 4 white, 14 blue, 6 green, and 1 yellow marbles in a bag. You put your hand into the bag and without looking, select a marble. What is the probability of removing the following?

a. P (white)

b. P (blue or green)

c. P (yellow)

d. P (not orange)

e. P (neither white nor blue)

Lesson #3 Experimental Probability:

The Experimental Probability is when you actually find the probability of an event in a real life experiment.

In the following experiment, you will find the experimental probability of the events in Example #1, Lesson #2. You are allowed to work in partners and you will be required to tally your results.

1. a. Refer to Example 1, lesson 2. What are the theoretical probabilities you found for the following:

i. The number 5

ii. An even number.

iii. A number greater than 4.

iv. Not the number 3.

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v. The number 9.

b. Roll a die 60 times. Record your results in the table below.

c. Refer to part (b) above. What are the experimental probabilities you obtained for the following? (note: your denominator will be 60) Express your answers as a fraction.

i. The number 5

ii. An even number.



v. The number 9.

d. Complete the following table comparing the theoretical probability to the experimental probability you obtained.

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2a. Roll the die an additional 60 times. Record your results in the table below.

b. Refer to part (a) above. What are the experimental probabilities you obtained this second time for the following? (once again your denominator is 60)

i. The number 5

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ii. An even number.



v. The number 9.

3. Use the data from both experiments to show the experimental probabilities for 120 rolls of the die. To find these probabilities, add the number of times each event occurred in the first and second experiments and divide by 120.

i. The number 5

ii. An even number.



v. The number 9.

4. Answer the following questions:

a. How closely did your experimental probability compare to the theoretical probability on the first 60 rolls.

b. How closely did your experimental probability compare to the theoretical probability on the secon 60 rolls.

c. How closely did your experimental probability compare to the theoretical probability for 120 rolls of the die.

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d. What would happen to these probabilities if you rolled the die 1000 times?

Lesson #4 Probability Continued

In this lesson we will try to solve the probability of more than one event.

Example 1 What is the probability of tossing each of the following with a penny, a nickel, and a dime?

a. P (3 Heads)b. P (2 Heads, 1 Tail)c. P (1 Head, 2 Tails)d. P (3 Tails)

Solution: Since the outcome of each coin does not depend on the outcome of the others, the probabilities are compounded independent events. We need to know the total number of possible events. Tree diagrams often help in this situation.

Step 1: To draw a tree diagram, begin with the possibilities of the first coin toss, that is Heads or Tails.

OUTCOMES

Coin #3 H HHH

Coin #2H T

HHT H HTH

Head T T HTT

Coin #1 H H THH

Tail T THT

T H TTH

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T TTT Step 1 Step 2 Step 3

Step 2: Next, indicate the possibilities of the second coin.

Step 3: Indicate the possibilities of the third coin.

Therefore, if you count all the outcomes in the tree diagram, you have 8. This is your denominator for the probability.

a. P (3 Heads) = 1/8

b. P (2 Heads, 1 Tail) = 3/8

c. P (1 Head, 2 Tails) = 3/8

d. P (3 Tails) = 1/8

Example #2 What is the probability of rolling each of the following with 2 dice?

a. P (even sum)b. P (odd sum)

For this example the best way to solve it is to create a table that shows all the outcomes.

Total Sums = 36. Total number of even sums = 18, odd sums = 18.

a. P (even sum) = 18/36 = ½ b. P (odd sum) = 18/36 = ½

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Assignment #4.

1. Toss one coin and spin the given spinner.

a. Draw a tree diagram and list all possible outcomes.

b. Find the following probabilities.

P (Heads, Red)

P (Tails, Blue)

P (Tails, not Green)

2. A jar contains the following 4 marbles: 1 blue, 1 green, 1 red, and 1 yellow. Students select two marbles from the jar. When the first marble is selected, it is returned to the jar before the second marble is selected.

a. Draw a tree diagram and list all possible outcomes.

b. Find the following probabilities:

P (blue, blue)

P (red, yellow)

P (neither marble is green)

P (two the same color)

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3. Two dice are rolled. The numbers displayed on each die are added. What is the probability of each of the following events? Refer to the table in example 2.

a. P (7)

b. P (12)

c. P (6 or 11)

d. P (> 10)

e. P (both dice display the same number)

f. P (sum is greater than 3 and less than 10)

4. Assume it is equally likely that a child is born a boy or a girl.

a. Draw a tree diagram and list the possible outcomes for a family of 2 children.

b. What is the probability that in ta family of 2 children, both will be girls.

c. Extend the tree diagram and list the possible outcomes for 3 children. What is the probability that in a family of 3, all will be boys?

5. A pair of dice are tossed. The results are multiplied with one another.

a. Draw a table to list all possible outcomes.

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b. What is P (even number)

c. What is P (≥ 20)

Lesson #5 Probability and Odds

The likelihood of an event occurring is not always expressed in terms of probability. The likelihood of an event occurring can also be expressed in terms of the odds in favor of it occurring. The probability of an event occurring and the odds in favor of it occurring are not the same thing.

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Probability = total correct events total possible events

Odds in favor = favorable outcomes : unfavorable outcomes

Odds against = unfavorable outcomes : favorable outcomes

Example 1

Three coins are tossed, determine:

a. The probability of 3 coins all landing heads.

b. The odds in favor of 3 coins all landing heads.

c. The odds against 3 coins all landing heads.

Example 2 Silas is buying new television for $750.

It has a 1 year warranty from the manufacturer The store sells a 3 yr extended warranty for $90 11 000 televisions did not need repairs over a 3 yr period and

1000 television did. a. What are the odds in favor of needing repairs?

b. What are the odds against needing repairs?

c. What is the probability that the television will need repairs during the 3 yr warranty.

Example 3

The odds in favor of an event occurring are 3:1. Determine the probability of the event occurring.

The odds of the event occurring are 3:1, so the number of ways the event can occur is 3 and the number of ways the event cannot occur is 1.

This means that the total number of possible events is 3 + 1 = 4.

The probability of the event occurring is 3 / 4 or 75%.

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Assignment #5

1. The odds against a hockey team winning a game are 2 : 9. What is the probability that the team will win a game?

2. Nathan plays basketball. He has scored on 2 out of 10 shots. He says that his odds against scoring are 4 to 1. Do you agree? Explain.

3. Each letter of the word MATHEMATICAL is written on a different card and placed face down on a table.

a. Determine the probability of drawing an “M”.

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b. Determine the odds in favor of drawing an “M”.

c. Determine the probability of not drawing an “M”.

d. Determine the odds against drawing an “M”.

4. Susan works at an appliance store in Brandon. The odds that a new vacuum cleaner will need repairs in the first 4 years are 1 : 3. What is the probability that a new vacuum cleaner will need repairs? Is this a good vacuum cleaner in your opinion? If the vacuum cleaner cost $350 and the manufacturer gave a 4 year warranty that cost $50, would you buy the warranty? Why or why not?

5. A die is rolled. Find the following:

a. The probability of rolling a number greater than two.

b. The odds in favor of rolling a number greater than two.

c. The probability of not rolling an even number.

d. The odds against rolling an even number.

6. In a class of 32 students, 18 students take an Art option, 10 other students take a Drama option, and the rest of the students take a Choir option. One student is selected at random. Find the following:

a. The odds in favor of the selected student taking Drama.

b. The odds against the selected student taking Choir.

c. The odds in favor of the selected student either taking Art or Choir.

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7. The Health Science Center Lottery states that players have a 1 in 12 chance of winning something in their lottery. What are the odds in favor and the odds against winning this lottery?

8. In roulette, the odds of winning on a single number is 1 : 37. What is the probability of winning in this game? What is the probability of not winning?

Lesson #6 Expected Value

Expected value is an application of probability which involves the likelihood of a gain or a loss. Expected value is relevant in business, insurance, and many situations in our daily life.

To determine the expected value of something, first you find the probability of the required event, and then you find the loss or gain associated with that event. You then multiply these together.

Example

Consider the following game. You have a 1 in 5 chance of winning the game and a 4 in 5 chance of losing the game. The game costs $1 to play each time. If you win the game, you receive $4 and if you lose the game, you receive nothing. Find the expected value of the game.

Solution:

There are two events: winning the game or losing the game. The probability of winning the game is 1/5 and the probability of losing the game is 4/5.

Event Probability Amount Won

Cost of Playing

PayoffProbability

XPayoff

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Winning the Game

Losing the Game

To find the expected value of the game, find the sum of the 2 probabilities multiplied by their payoffs.

Expected Value =

Assignment #6

1. Mr. Krahn is at the Red River Ex. He wants to play the ring toss and win a stuffed animal for his wife. It will cost $3 each time he plays the game. He has a 1 in 10 chance of winning the $25 stuffed toy, a 1 in 5 chance of winning $5 stuffed toy, and a 7 in 10 chance of winning nothing.

Fill in the following chart below.

Event ProbabilityAmount

WonCost of Playing Payoff

Probability X

Payoff

What is the expected value of the game?

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2. Aaron is thinking about adding collision insurance when he renews his car-insurance policy next year. This will increase the cost of his insurance by $720 / yr.

Statistically, there is a 99.4% chance that Aaron will not have a collision in the next year.

If Aaron has a collision, the insurance company will pay $5000.

a. What is the probability that Aaron will have a collision?

b. What is the expected gain for the company if Aaron adds collision insurance?

Expected gain = (probability of gain)(gain amount)–(probability of loss)(loss amount)

c. Should Aaron add collision insurance? Explain?

3. A building contractor sets her probability of winning a contract at .30. The contract is worth $25 000 and she determines it will cost her $2400 to prepare a contract proposal.

a. Find the expected value of the contract proposal.

b. Is it a good financial decision for her to bid on the contract? Why?

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c. What other factors might she consider before deciding whether to bid on the contract?

4. Dan is buying a clothes dryer that costs $825.

The clothes dryer has a 1 yr warranty from the manufacturer The odds against needing repairs over the 5 yrs are 3:17.

a. What is the probability that the clothes dryer will need repairs during the extended – warranty period?

b. What is the probability that the dryer will not need repairs.

c. The company estimates that it costs $200 to repair a dryer. Suppose that Dan buys the warranty. What is the expected gain for the store?

d. Should Dan buy the warranty? Explain. What assumptions did you make?

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