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Chapter 4

Chapter 4. Probability. Definitions. A probability experiment is a chance process that leads to well-defined results called outcomes. An outcome is the result of a single trial of a probability experiment. A sample space is the set of all possible outcomes of a probability experiment.

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Chapter 4

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  1. Chapter 4 Probability

  2. Definitions • A probability experiment is a chance process that leads to well-defined results called outcomes. • An outcome is the result of a single trial of a probability experiment. • A sample space is the set of all possible outcomes of a probability experiment.

  3. The probability of any event E is

  4. The probability of an event is a number (either a fraction or a decimal) between zero and one, inclusive.

  5. Cardinal Rule • Never Count Anything Twice!!!

  6. If the probability of an event is zero, the event is impossible. • If the probability of an event is one, the event is certain. • The sum of the probabilities of the outcomes in the sample space is one.

  7. Cardinal Rule • Never Count Anything Twice!!!

  8. Example 1 • Roll a fair die. • Find the sample space.

  9. Example 2 • A family wants to have three children. • How many elements are in the sample space? • List the elements in the sample space.

  10. Example 1 • Roll a fair die. • Find the sample space. • Find the probability of rolling an even number. • Find the probability of rolling a number that is prime.

  11. Example 2 • A family wants to have three children. • How many elements are in the sample space? • List the elements in the sample space. • Find the probability that the family has one girl. • Find the probability that the family has at most one girl. • Find the probability that the family has at least one boy.

  12. Example 3 • A card is drawn from an ordinary deck of cards. • Find the probability of getting a jack. • Find the probability of getting the 6 of clubs. • Find the probability of getting a red card. • Find the probability of getting a red jack. • Find the probability of getting a red card or a jack. • Find the probability of getting a jack and a king.

  13. Cardinal Rule • Never Count Anything Twice!!!

  14. Two events are mutually exclusive if they cannot occur at the same time.

  15. Example 4 • A jar contains 3 red, 2 blue, 4 green, and 1 white marble. A marble is chosen randomly. • Find the probability of choosing a red marble. • Find the probability of choosing a red or a blue marble. • Find the probability of choosing a red and a blue marble.

  16. Example 5 • A jar contains 3 red, 2 blue, 4 green, and 1 white marble. Two marbles are randomly chosen, choosing the 2nd marble after replacing the first. • Find the probability of choosing 2 red marbles. • Find the probability of choosing a red and then a green marble. • Find the probability of choosing a red and a green marble. • Find the probability of choosing two white marbles.

  17. Two events A and B are independent if the fact that A occurs does not affect the probability of B occurring.

  18. Example 6 • A jar contains 3 red, 2 blue, 4 green, and 1 white marble. Two marbles are randomly chosen, choosing the 2nd marble without replacing the first. • Find the probability of choosing 2 red marbles. • Find the probability of choosing a red marble and then a green marble. • Find the probability of choosing two white marbles.

  19. When the outcome or occurrence of the first events affects the outcome or occurrence of the second event in such a way that the probability is changes, the events are said to be dependent events.

  20. Example 7 • If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that two randomly selected federal prison inmates will not be U. S. citizens.

  21. Example 8 • A flashlight has 6 batteries, two of which are defective. If two batteries are randomly selected, find the probability that both are defective.

  22. Example 9 • In a large shopping mall, a marketing agency conducted a survey on credit cards. The results are below.

  23. Cardinal Rule • Never Count Anything Twice!!!

  24. If one person is randomly selected, • Find the probability that the person owns a credit card. • Find the probability that the person is employed. • Find the probability that the person owns a credit card and is employed. • Find the probability that the person owns a credit card or is employed. • Find the probability that the person owns a credit card given that the person is employed. • Find the probability that the person is employed given that the person owns a credit card.

  25. If you see the word “given” in a problem, the sample space has changed. • The new sample space is what comes after the word “given” • In other words, in our formula for probability, the denominator changes to what comes after the word “given”

  26. Cardinal Rule • Never Count Anything Twice!!!

  27. Are the events “owning a credit card” and “being employed” mutually exclusive events? Why or why not?

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