2. Probability: Living with the Odds
7D Discussion Paragraph
1 web
32. Cancer vs. Heart Disease
33. Travel Safety
34. Life Expectancy
1 world
35. Travel Safety
36. Vital Statistics
37. Life Expectancies
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3. Counting and Probability
Unit 7E
Counting and Probability
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4. Arrangements with Repetition
If we make r selections from a group of n choices, a total of different arrangements are possible.
Example: How many 7-number license plates are possible?
Current debates over issues such as the increase in area codes, toll-free numbers, and even the challenge of Social Security numbers in the future are great topics to bring into this discussion.
There are 10 million different possible license plates.
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5. Arrangements with Repetition CN (1)
a. How many seven-symbol license plates are possible if both numerals and letters can be used in any order?
b. How many six-character passwords can be made by combining lowercase letters, uppercase letters, numerals, and the $, and &?
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6. Permutations e.g., ABCD is different from DCBA
We are dealing with permutations whenever
all selections come from a single group of items,
no item may be selected more than once,
and the order of arrangement matters.
e.g., ABCD is different from DCBA
The total number of permutations possible with a group of n items is n!, where
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7. Class Schedules CN (2)
A middle school principal needs to schedule six different classes—algebra, English, history, Spanish, science, and gym—in six different time periods.
2. How many different class schedules are possible?
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8. The Permutations Formula
If we make r selections from a group of n choices, the number of permutations (arrangements in which order matters) is
Example: On a team of 10 swimmers, how many possible 4-person relay teams are there?
Contrast with the combination slide to emphasize that if order matters we use a permutation.
There are possible relay teams!
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9. The Permutations Formula
Example: If an international track event has 8 athletes participating and three medals (gold, silver and bronze) are to be awarded, how many different orderings of the top three athletes are possible?
Contrast with the combination slide to emphasize that if order matters we use a permutation.
There are 336 different orderings of the top three athletes!
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10. Leadership Election CN (3)
A city has 12 candidates running for three leadership positions. The top vote-getter will become the mayor, the second vote-getter will become the deputy mayor, and the third vote-getter will become the treasurer.
3. How many outcomes are possible for the three leadership positions?
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11. Batting Orders CN (4)
4. How many ways can the manager of a baseball team form a (9-player) batting order from a roster of 15 players?
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12. Combinations e.g., ABCD is considered the same as DCBA
Combinations occur whenever
all selections come from a single group of items,
no item may be selected more than once,
and the order of arrangement does not matter
e.g., ABCD is considered the same as DCBA
If we make r selections from a group of n items, the number of possible combinations is
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13. The Combinations Formula
Example: If a committee of 3 people are needed out of 8 possible candidates and there is not any distinction between committee members, how many possible committees would there be?
Contrast with the previous slide to emphasize that if order doesn’t matter we use a combination. A simple example with letters ABCD to fit in two boxes provides an opportunity to see where the division by r! comes from.
There are 56 possible committees!
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14. Ice Cream Combinations CN (5)
Suppose that you select 3 different flavors of ice cream in a shop that carries 12 flavors.
5. How many flavor combinations are possible?
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15. Poker Hands CN (6a-b)
a. How many different five-card poker hands can be dealt from a standard deck of 52 cards?
b. What is the probability of one particular hand, such as a royal flush of hearts (ace, king, queen, jack, and 10)?
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16. (Not) Winning the Lottery CN (7)
Suppose you play the lottery in which the winner is chosen by drawing 6 balls at random from a drum containing 52 numbered balls (numbered 1 through 52).
7. What is the probability that your 6 numbers will match the 6 winning numbers?
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17. Probability and Coincidence CN (8)
Coincidences are bound to happen.
Although a particular outcome may be highly unlikely, some similar outcome may be extremely likely or even certain to occur.
8. What is the probability that at least two people in a class of 25 have the same birthday?
The probability graph makes for a great lead in to logistic curves studied in section 8-C. To be 50% confident that there are two people or more who share some common birthday you only need around 23 people in the group. This assumes that birthdays are randomly scattered throughout the calendar year and it disregards leap year. To be 99% confident requires about 57 people in the group. This is always a fun class experiment to do when you have more than 30 in the room. It’s nice to open with the 50% question first and many students will suggest that 65/2 or about 183 people would be needed to be 50% confident. This is another example where intuition and probability sometimes tend to part ways.
The answer has the form
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18. Birthday Coincidence
The probability that all 25 students have different birthdays is
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19. Birthday Coincidence
The probability that at least two people in a class of 25 have the same birthday is
P(at least one pair of shared birthdays)
= 1 – P(no shared birthdays)
≈ 1 – ≈ ≈ 57%
This is always a fun class experiment to do when you have more than 30 in the room. It’s nice to open with the 50% question first and many students will suggest that 65/2 or about 183 people would be needed to be 50% confident. This is another example where intuition and probability sometimes tend to part ways. To be 50% confident that there are two people or more who share some common birthday you only need around 23 people in the group. This assumes that birthdays are randomly scattered throughout the calendar year and it disregards leap year. To be 99% confident requires about 57 people in the group.
The probability that at least two people in a class of 25 have the same birthday is approximately 57%!
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20. Birthday Coincidence
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room? 1
y = Probabilities
x = People in Room
The probability graph makes for a great lead in to logistic curves studied in section 8-C.
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21. Birthday Coincidence
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room? 1
y = Probabilities
x = People in Room
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22. Birthday Coincidence
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room? 1
y = Probabilities
x = People in Room
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23. Birthday Coincidence
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room? 1
y = Probabilities
x = People in Room
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24. Birthday Coincidence
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room? 1
y = Probabilities
x = People in Room
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25. Birthday Coincidence
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room? 1
y = Probabilities
x = People in Room
How many people in the room would be required for 100% certainty?
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26. Quick Quiz CN (9)
9. Please answer the 10 quick quiz multiple choice questions on p. 465.
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27. Homework 7E Discussion Paragraph 7D Class Notes 1-9 P.466:1-10 1 web
The “Monty Hall” Problem
1 world
59. Lottery Chances
60. Amazing Coincidence
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