2. Probability: Living with the Odds
6D Discussion Paragraph
1 web
50. Vital Significance
51. Recent Polls
52. Genetically Modified Foods
1 world
53. Statistical Significance
54. Significant Experiment?
55. Margin of Error
56. Confidence Intervals
57. Hypothesis Testing
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3. Fundamentals of Probability
Unit 7A
Fundamentals of Probability
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4. Lotteries
Lotteries are big business and a major revenue source for many governments. By 2009, all but seven states in the US had some type of lottery, generating a national total of more than $52 billion in sales.
About 1/3 of that money ($17 billion) ends up as state revenue, with the rest going to prizes and expenses.
Lotteries present many lessons in probability, and lottery stats can fuel great debate over whether lotteries are an appropriate way for governments to generate revenue.
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5. Definitions
Outcomes are the most basic possible results of observations or experiments.
An event consists of one or more outcomes that share a property of interest.
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6. Expressing Probability
The probability of an event, expressed as P(event), is always between 0 and 1 (inclusive).
A probability of 0 means the event is impossible and a probability of 1 means the event is certain. 1
0.5
Certain
Likely
Unlikely
50-50 Chance
Impossible
0 ≤ P(A) ≤ 1
Meteorology offers a rich source of ideas here. Would you take an umbrella to work if the chance for rain is at 10%? How about 90%?
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7. Theoretical Method for Equally Likely Outcomes
Step 1: Count the total number of possible outcomes.
Step 2: Among all the possible outcomes, count the number of ways the event of interest, A, can occur.
Step 3: Determine the probability, P(A).
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8. Outcomes and Events
Assuming equal chance of having a boy or girl at birth, what is the probability of having two girls and two boys in a family of four children?
Of the 16 possible outcomes, 6 have the event two girls and two boys.
P(2 girls) = 6/16 = 0.357
One of the challenges many students have with probability is knowing when to trust intuition and when to back away from it. Many students would answer the question with a 50%. This problem is a classic example of the importance of looking at the total number of outcomes.
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9. Playing Card Probabilities CN (1)
There are 52 cards in a standard deck. There are four suits, hearts, diamonds, spades and clubs. Each suit has card for the numbers 2 through 10 plus a jack, queen, king, and ace (for a total of 13 cards in each suit). Notice that 2 suits are red and 2 suits are black.
1. If you draw one card from a standard deck, what is the probability that it is a spade?
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10. Guessing Birthdays CN (2)
You select a person at random from a large conference group.
2. What is the probability that the person has a birthday in July?
Assume 365 days a year.
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11. Two Boys and a Girl CN (3)
3. What is the probability that a family with three children has two boys and one girl?
Assume boys and girls are equally likely.
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12. Empirical and Subjective Probabilities
An empirical probability is based on observations or experiments. It is the relative frequency of the event of interest.
A subjective probability is an estimate based on experience or intuition.
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13. 500 Year Flood CN (4)
Geological records indicate that a river has crested above a particular high flood level just four times in the past 2000 years.
What is the empirical probability that the river will crest above this flood level next year?
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14. Empirical Coin Tossing CN (5)
If you are interested only in the number of heads in coin tosses, then the possible events when tossing two coins are 0 heads, 1 head, and 2 heads. Suppose you repeat a two-coin toss 100 times and your results are as follows.
0 heads occurs 22 times
1 head occurs 51 times
2 heads occurs 27 times
Compare the empirical probabilities to the theoretical probabilities.
5. Do you have reason to suspect that the coins are unfair?
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15. Three Types of Probabilities
Theoretical probability
The chance of
rolling a 4 is 1 out of 6.
Empirical probability
Subjective probability
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16. Three Types of Probabilities
Theoretical probability
She’s a 92% free throw
shooter for the season.
Empirical probability
Subjective probability
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17. Three Types of Probabilities
Theoretical probability
There’s about a 70%
chance she will go out
on a date with me.
Empirical probability
Subjective probability
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18. Which Type of Probability? CN (6a-c)
6. Identify the method that resulted in the following statements:
a. I’m 100% certain that you’ll be happy with this car.
b. Based on data from the Department of Transportation, the chance of dying in an automobile accident during a one-year period is 1 in 7000a
c. The chance of rolling a 7 with a twelve-sided die is 1/12.
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19. Probability of an Event Not Occurring
If the probability of an event A is P(A), then the probability that event A does not occur is 1 – P(A).
Since the probability of a family of four children having two girls and two boys is 0.375, what is the probability of a family of four children not having two girls and two boys?
P(not 2 girls) = 1 – = 0.625
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20. Not Two Boys CN (7)
7. What is the probability that a randomly chosen family with three children does not have two boys and one girl?
Assume boys and girls are equally likely.
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21. Making a Probability Distribution
A probability distribution represents the
probabilities of all possible events. To make a
probability distribution, do the following:
Step 1: List all possible outcomes. Use a table or figure if it is helpful.
Step 2: Identify outcomes that represent the same event and determine the probability of each event.
Step 3: Make a table listing each event and probability.
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22. A Probability Distribution
All possible outcomes and a probability distribution for the sum when two dice are rolled are shown below.
Possible outcomes
Point out that the reason 7 is considered a lucky number in many gambling games is simply because there are more ways to roll a 7 than any other number.
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23. Tossing Three Coins CN (8)
8. Make a table of the probability distribution for the number of heads that occur when three coins are tossed simultaneously.
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24. Two Dice Distribution CN (9)
Make a probability distribution for the sum of the dice when two dice are rolled.
9. What is the most probable sum?
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25. Odds
Odds are the ratio of the probability that a particular event will occur to the probability that it will not occur.
The odds for an event A are
The odds against an event A are
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26. Two Coin Odds CN (10a-b)
10a. What are the odds for getting two heads in tossing two coins?
b. What are the odds against it?
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27. Horse Race Payoff CN (11)
At a horse race, the odds on Blue Moon are given as 7 to 2.
11. If you bet $10 and Blue Moon wins, how much will you gain?
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28. Quick Quiz CN (13)
13. Please answer the 10 quick quiz multiple choice questions on p.425.
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29. Homework 7A Discussion Paragraph 6D P. 425-6: 1-12 1 web
70. Blood Groups
71. Accidents
1 world
72. Probability in the News
73. Probability in your Life
74. Gambling Odds
Class Notes 1-13
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