Math 3202 Probability and Odds - · PDF fileMath 3202 Section 1.2 Probability and Odds Notes 1...

Math 3202 Section 1.2 Probability and Odds Notes

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Math 3202 Probability and Odds

Activate Prior Knowledge: Fractions, Decimals and PercentagesComplete the following table showing equivalent value for the fractions, decimals, and percentages. Remember to always leave Fractions in lowest terms!

Fraction Decimal Percentage

15

0.7012%

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Section 1.2 Probability and Odds

Have you heard expressions like the following?

* The home team was the odds on favourite to win the trophy.

* The odds were good that the job would be completed on time.

* Brent wants to be picked for the Olympic swim team. What are the odds of that?What are the odds in favour of drawing the queen of spades from a wellshuffleddeck of cards?

To calculate the odds of an event occurring, compare it to thenumber of alternative events.

A deck of cards contains 1 queen of spades and 51 other cards. The odds infavour of drawing the queen of spades can be stated as the ratio 1:51, or one tofiftyone, because there are 52 cards in a deck.

Compare this to a statement of probability. What is the probability that you will draw the queen of spade from a deck of cards? The probability can be stated as one in fiftytwo.

Probability compares an event to all the possible outcomes.

The odds compare an event against the alternatives.

You may wish to find the odds against an outcome. In this case, the ratio compares the number of possible unfavourable outcomes to the outcome you wish to occur. The odds against drawing the queen of spades from a deck of cards is fiftyone to one, or 51:1


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Example 1

A television station is hosting a contest. Alice, a participant, can choose 1 of 7 doors to open. She will win whatever prize is behind the door. Behind the 7 possible doors are the following prizes.

* one offers a trip

* two contain $20 gift certificates

* three award restaurant dinners

* one has not prize.

a) What are the odds of choosing the door with no prize?

* one offers a trip

* two contain $20 gift certificates

* three award restaurant dinners

* one has not prize.

b) What is the probability of choosing the door with no prize?

c) What are the odds of Alice winning a gift certificate?


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Example 2

A tire manufacturing company does random testing of tires coming off the production lines to ensure that they are being produced correctly. After testing, the quality control manager calculates that the experimental probability of a tire having a defect is 0.003.

a) What are the odds in favour of a tire being defection?

b) What are the odds against a tire being defective?

c) What are the odds in favour of a tire being correctly manufactured?

d) In a production run of 30 000 tires, how many tires would you expect tobe defective? Nondefective?

Mental Math and Estimation

Sophia runs a summer camp for children at a local community centre. At the end of the summer, she sends a survey to parents to find out which activities she should offer the next summer. The table shows the survey results.

Estimate the odds in favour of a child choosing the music class if all the activities were offered on the same day.

Activity Votes

Trip to Water Park 34

Hike to Lake 29

Bicycle ride through Park 21

Crafts 11

Music Class 9


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Assignment Section 1.2

Name: ________________


1. Ginger is a cook at Klondike Rib & Salmon Barbecue in Whitehorse, YT. Sheis cooking a meal for a culinary competition. She has the following bottles of spices on her spice rack;

* 1 bottle of thyme * 1 bottle of oregano * 1 bottle of cinnamon

* 1 bottle of cilantro * 2 bottles of pepper * 2 bottles of paprika

* 3 bottles of her secret mixture of spices * 1 bottle of nutmeg

If she randomly picks a spice bottle off the rack without looking:

a) What is the probability that Ginger will choose cilantro?

b) What is the probability that Ginger will choose pepper?

c) What are the odds that Ginger will choose paprika?

d) What are the odds that Ginger will choose the correct spice if her reciperequires her secret mixture of spices?

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2. A toolbox contains 2 sizes of Robertson screwdrivers, 4 sizes of Phillipsscrewdrivers, and 3 sizes of flathead screwdrivers. If you randomly pull out a screwdriver:

a) What are the odds in favour of it being a Robertson screwdriver?

b) What are the odds against it being a flathead screwdriver?

c) What are the odds in favour of it being either a Phillips or a Robertsonscrewdriver?

3. Complete the chart below:

Event Odds in Favour Odds Against Probability

Drawing a 10 from a deck of cards

Rolling a sum of 6 using two dice

Drawing a red card from a deck of cards

Rolling triples with three dice

Rolling a sum of 2 with two dice

Odds and Probability


a) 2: 7

b) 2:1

c) 2:1

a) 1 out of 12, 0.083 or 8.3%

b) 1 out of 6, 0.17 or 17%

c) 1:5

d) 1:3

1. 2.

3.

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Section 1.3 Experimental Probability

Have you heard any of these statements?

* There is a 85% of snow.

* three out of four dentists recommend a certainbrand of toothpaste

* Goalie Roberto Luongo's saves statistic was 0.927 at the 2010 Olympic WinterGames in Vancouver.

* There is a chance that a medication will produce side effects.

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Probability is a prediction of how likely an event is to occur. It can be stated as a decimal, a percentage, a fraction, or in words, as shown in the examples above.

Math 3202 Section 1.3 Experimential and Theoretical Probability Notes

A statement of probability compares the number of times an event of interest occurs to the total number of possible outcomes. The formula for calculating the probability (P) of event A is:

P (A) = number of occurrences of event A total number of possible outcomes

In the examples above, the probability is based on measuring a sample of data or running a trial to find out what the probability is. The is known as experimental probability , because it is based on known data. Using past results allows you to extrapolate and make a prediction about the likelihood of future events.

The more accurate the data used to calculate the probability, the better decisions you will make based on the probability.

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Example 1

Shelley is the production manager of an assembly line that makes compact fluorescent light bulbs. She received a complaint from a store that a shipment contained a large number of defective bulbs. Shelley decided to test how probable it is that the light bulbs from the assembly line are defective.Shelley has the workers on the assembly line randomly choose light bulbs from the assembly line and test them. Out of 265 light bulbs tested in the random sample, 2 were defective.

a) Express the probability of a light bulb being defective as a decimal,apercentage, a fraction, and in a statement


b) In a shipment of 5000, how many light bulbs are likely to be defective?

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Example 2Georgina bought a bag of bulk candies. She counted them and found that 15 were red, 10 were green, 12 were purple, and 13 were orange.

a) Calculate the probability of a randomly chosen candy from thebag being red. Express the probability as a fraction, a decimal, a percentage, and in words.


b) What is the probability of a randomly chosen candy being green orpurple?

c) If the distribution of colors in a bag of 1000 candies is the same, howmany would Georgina expect to be red?

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a) P (Red Candy) = 3 , 0.3, 30% or 3 out of 10 10

b) P(green or purple) = 11, 0.44, 44% or 11 out of 25 44

c) 300 are likely to be red


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Assignment: Experimential Probability

Name: _______________

1. Alexie is a veterinary assistant. She assisted in the delivery ofDalmatian puppies, and found out that one of the puppies was born deaf. She did research and found a study that tested 2225 Dalmatian puppies. The study found that the probability of puppies being born deaf was 2 out of 25. How many puppies in the study were born deaf?


2. The chief mechanic at a car dealership in Quesnel, BC, knowsthe probability of a particular model of truck having problems with the transmission is 2.4%. If the dealership has sold 3250 of these trucks, how many can the dealership expect to come in for repair?

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3. Everett, a school cafeteria cook, has been asked to change the menu to offerhealthier choices for the students and staff. Everett has created 5 different dishes for 300 students to sample and vote for their favorite.

a) Based on the results of the sampling, calculate the probability that a studentwould choose each meal. Express the probability as a fraction and as a percent.

b) If the school has 1348 students, how many of each meal would Everett have tomake each week (based on a 5day school week)? How does this information help Everett with the work he does?

Meal Number who voted for the meal

Probability of choice as a fraction

Probability of choice as a percent

Number of Meals to be made in 5 days

Roast Beef and Potatoes 78

Egg Salad Sandwich 22

Homemade Chicken Veg Soup 36

Vegetable Fajitas 83

Chili 81


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Section 1.2 Theoretical Probability

Probability can be calculated without doing an experiment when their is a finite number of possible outcomes. When you toss a sixsided die, how many possible outcomes are there?

If you draw a card from a deck of cards, what is the probability that you will draw an ace?

In both these cases, there is a finite number of possible outcomes.


Probability of Drawing an Ace = four possible eventstotal number of possible outcomes

P = 452

Simplify

P = 113

There is a 1 in 13 chance of drawing an ace. Calculating the probability of an event occurring when there is a limited number of possible outcomes is known as theoretical probability.

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Example 1

Use the spinner shown below to calculate the following probabilities.

a) What is the probability that thespinner will stop on green?

b)What is the probability that the

spinner will not stop on yellow?

c) What is the probability that the spinner will stop on red or orange?


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Example 2

A bag of finishing nails contains 5 nails measuring 1", 5 nails measuring 2", and 7 nails measuring 3". A carpenter randomly takes out one nail then, without replacing that nail, takes out a second nail.

a) What is the probability that the first nail will be a 1" nail?

b) If the first nail chosen was a 1" nail, what is the probability that thesecond nail will be a 3" nail?


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Assignment: Theoretical Probability

Name: _________________


1. a) What is the theoretical probability of rolling doubles with two dice?

b) Zoe decided to test this probability. She rolled the dice 20 times, andgot doubles on 4 of the rolls. What is the experimental probability of rolling doubles?

c) Using Zoe's experimental results, what do you think would havehappened if she rolled the dice 500 times?

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d) If the other 20 students in Zoe's math class all tried this experimentas well, and each person rolled the dice 75 times, how do you think the results would compare to the theoretical probability? Explain your reasoning.


2. Daniel volunteered to help with the campaign of a candidaterunning to be a Member of Parliament in his riding. Daniel randomly called 500 households in the riding to poll people's voting intentions. Daniel obtained the following responses to 4 options.

Candidate Votes

Bill 218

Joyce 102

Mansour 89

Undecided 91

a) What is the theoretical probability that a person Daniel spoke towould select Joyce?

b) What is the experimental probability that a person Daniel spoke towould vote for Bill? Write your answer as a percent.

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c) If a person walking down the street was randomly selected, what isthe probability that he or she would be planning to vote for Mansour? Express your answer as a decimal. Would you use an experimental or theoretical calculation of the probability? What other factors might you need to consider when predicting the person's response? Explain your reasoning.

3. A bowl of fruit contains 6 apples and 4 oranges. Olivier randomlychooses one fruit from the bowl and eats it, then randomly chooses a second piece of fruit from the bowl.

a) What is the probability that the first piece of fruit is an apple?

b) If the first piece of fruit was an apple, what is the probability that thesecond piece of fruit is also an apple?

8. Toivo has 200 songs on his MP3 player. He likes several different kidsof music, as shown in his music collection: 96 of the songs are hip hop, 34 are alternative rock, 28 are rock, 12 are classical, and 30 are dance music.

a) If the song selection is random, what is the probability that the firstsong he plays will be a hip hop song?

b) If the song selection is random, what is the probability that the firstsong he plays will be an alternative rock or a rock song?


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a) 3 out of 5, 0.6, or 60%

b) 5 out of 9, 0.56, or 56%

a) 12 out of 25, 0.48, or 48%

b) 31 out of 100, 0.31 or 31%


a) 1 out of 4

b) 43.6%

c) Answers will vary. The experimentalprobability is 0.178

a) 1 out of 6b) 0.2, 20%, 2 out of 10 orc) Zoe would roll doubles 100 times.d) Answers will vary.If the probability shown in Zoe's experiment in part a)is correct, the students will roll doubles about 300 times.

According the theoretical probability, the students wouldroll doubles about 250 times.

Actual number of rolls of doubles would probably be closerto the theoretical probability than the experimental probability.

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