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Math 3370 7.3: Concepts Related to Chance

Chapter 7

Definitions:

1. A numerical value the represents the degree to which we believe that event will or will not happen is the probability. number of favorable outcomes 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝐸 𝑛(𝐸) 𝑃(𝐸) = 𝑃(𝐸) = = number of total outcomes 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑆 𝑛(𝑆) Where 𝑛(𝐸) means the number of elements in 𝐸. 2. Each outcome to an experiment is an element of the sample space. The subset of a sample space is called an event. 3. The probability of an event is equal to the sum of the probabilities of all outcomes in that event. 4. An impossible event is an event that is not likely to happen. The probability of an impossible event is 0. 5. A certain event is an event that is most likely to happen. The probability of a certain event is 1. 6. The process of simulating a situation and then collecting and analyzing a large amount of data is called simulation, the Monte Carlo method. 7. The Law of Large Numbers states that as we perform an experiment over and over, the experimental probability will converge on a fixed number.

8.

If events 𝐴 and 𝐵 are mutually exclusive, then 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵).

𝐴

𝐵

9. If we have two events such that event 𝐴 and event 𝐵 are complementary, then 𝑃(𝐴) + 𝑃(𝐵) = 1.

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10. If the probability of event 𝐴 is 𝑥 and the probability of event 𝐵 is 𝑦, then the probability of the event “A then B” is 𝑥 ⋅ 𝑦.

11. Two events are dependent if the probability of the second event is affected by the outcome of the first event.

12. Two events are independent if the probability of the second event is not affected at all by the outcome of the first event.

13. The expected value is mean of a random variable. What you should anticipate happening in the long run of many trials of a game of chance. Calculated by finding the sum of multiplying the outcome * probability for each element.

7.4: Counting and Chance

Definitions:

14. Trees are used to calculate the probability of multiple events happening. The first set of branches is the first event happening, the second is the second event, and so on.

15. The number of different subsets of size 𝑟 that we can make from a set of size 𝑛 is the number of permutations of 𝑛 things taken 𝑟 at a time. We are talking about situations where order matters. 𝑛!

Generalization: 𝑛𝑃𝑟 = 𝑛(𝑛 − 1) ⋯ (𝑛 − 𝑟 + 1) = (𝑛−𝑟)!

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16. The combination of 𝑛 things taken 𝑟 at a time without or with repetitions. We are talking about situations where order does not matter. 𝑛!

Generalization: 𝑛𝐶𝑟 = 𝐶(𝑛, 𝑟) = 𝑟!(𝑛−𝑟)! =

𝑛(𝑛−1)⋯(𝑛−𝑟+1)(𝑛−𝑟)(𝑛−𝑟−1)! 𝑟!(𝑛−𝑟)(𝑛−𝑟−1)!

=

𝑛(𝑛−1)⋯(𝑛−𝑟+1) 𝑟!

Examples: Homework 2

1) In a class, 22 people live on campus and 2 people live off-campus. If we draw one student’s name from a hat, what is the probability that this student is living off-campus? (Enter your answer as a fraction.)

2) Peppermint Patty is very discouraged about her chance on a 10-item true-false quiz. What is her probability of getting a grade of at least 50% on the following? (Enter an exact number as an integer, fraction, or decimal.) (a) Two-item quiz

(a) Five-item quiz

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3) A gumball machine has 6 red gumballs and 6 green ones. If you buy two gumballs, what is the probability that both will be red? (Enter your answer as a fraction.)

4) In Chapter 8, you will be introduced to regular polyhedrons. Usually, dice shaped like cubes, but any of the regular polyhedrons could be used. (Enter both answers as a single fraction.) (b) The faces of 2 dodecahedrons are numbered 1 through 12. What is the probability of rolling a sum of 6 on 2 dodecahedral dice?

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5) A survey of 300 community college students yielded the following results. Democrat Republican Other 84 40 30 Freshman 67 64 15 Sophomores

On the basis of this survey, what is the probability of each of the following? a) A college student is a Republican. (Round your answer to three decimal places.)

b) A freshman is a Democrat. (Round your answer to three decimal places.)

c) A sophomore is a Democrat. (Round your answer to three decimal places.)

6) A school has a raffle for the 160 students in the fifth and sixth grades. There are 92 fifth graders. What is the probability that the grand-prize winner selected at random from 160 students will be a sixth grader? (Round your answer to three decimal places.)

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7) Janine’s boss has allowed her to have a flexible schedule. Her boss says she can pick whatever 5 days a week she will work. a) How many different work combinations can be made?

b) How many choices give her consecutive days off?

c) How many choices give her Wednesday off?

8) A pizza parlor offers the following toppings: mushrooms, peppers, broccoli, shrimp, and pepperoni. How many different kinds of large pizza (with tomato and cheese) are there? (Include a pizza with no toppings.)

9) What is the probability that a family with 8 children has 8 girls? (Give your answer as a fraction.)

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10) You have a job working for a mathematician. She pays you each day according to what card you select from a bag. Five of the cards say $240, two of them say $120, and three of them say $60. What is your expected (average) daily pay?

11) (a) Suppose you buy $20,000 of air travel insurance against death. If the probability of your dying on the flight is 1 in 300,000, what is the mean payoff for your insurance policy? (Round your answer to the nearest cent.)

(b) If the insurance cost $1, what is the average gain or loss per policy for consumers? (Round your answer to the nearest cent.)