1. Counting and Probability
Unit 7E
Counting and Probability
Ms. Young

2. 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.
Ms. Young

3. 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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4. 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!
Ms. Young

5. 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!
Ms. Young

6. 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
Ms. Young

7. 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!
Ms. Young

8. Probability and Coincidence
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.
Example: 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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9. Birthday Coincidence
The probability that all 25 students have different birthdays is
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10. 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%!
Ms. Young

11. 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.
Ms. Young

12. 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
Ms. Young

13. 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
Ms. Young

14. 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
Ms. Young

15. 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
Ms. Young

16. 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?
Ms. Young