Probability

1. Probability What Are the Chances?

2. Section 1The Basics of Probability TheoryObjectives: • Be able to calculate probabilities by counting the outcomes in a sample space. • Use counting formulas to compute probabilities. • Understand how probability theory is used in genetics. • Understand the relationship between probability and odds.

3. The Sample Space Random phenomenaare occurrences that vary from day-to-day and case-to-case. Although we never know exactly how a random phenomenon will turn out, we can often calculate a number called a probabilitythat it will occur in a certain way.

4. Sample Space cont. (2) • Example: Determine a sample space for the experiment of selecting an iPhone from a production line and determining whether it is defective. Solution: This sample space is { defective, nondefective }. • Example: Determine the sample space for the experiment. We roll two dice and observe the pair of numbers showing on the top faces. Solution: The sample space consists of the 36 listed pairs.

5. Sample Space cont. (3) • Example: Determine the sample space for the experiment. Three children are born to a family and we note the birth order with respect to gender. Solution: We use the tree diagram to find the sample space: {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}.

6. Events Example: Write each event as a subset of the sample space. a) A head occurs when we flip a single coin. b) Two girls and one boy are born to a family. c) A sum of five occurs on a pair of dice. Solution: a) The set {head} is the event. b) The event is {bgg, gbg, ggb}. c) The event is {(1, 4), (2, 3), (3, 2), (4, 1)}.

7. Probability Experimental (empirical) Probability

8. Experimental Probability Example: A pharmaceutical company is testing a new drug. The company injected 100 patients and obtained the information shown. Based on the table, if a person is injected with this drug, what is the probability that the patient will develop severe side effects? Solution: We obtain the following probability based on previous observations.

9. Theoretical Probability Example: The table summarizes the marital status of men and women (in thousands) in the United States in 2006. If we randomly pick a male, what is the probability that he is divorced? Note: If we “randomly pick” a male, then any one of the males is equally likely to be picked.

10. Theoretical Probability cont. (2) Solution: We are only selecting a male, so we consider our sample space to be the 60,955 + 2,908 + 10,818 + 2,210 + 39,435 = 116,326 males. The event, call it D, is the set of 10,818 men who are divorced. Therefore, the probability that we would select a divorced male is

11. Counting and Probability • Probabilities may be based on empirical information. For example, the result of experimental data. • Probabilities may also be based on theoretical information such as combination formulas. • Theoretical probability example: We flip three fair coins. What is the probability of each outcome in this sample space? Solution: Since the eight outcomes are equally likely, each has a probability of .

12. Counting and Probability cont. (2) Example: We draw a 5-card hand randomly from a standard 52-card deck. What is the probability that we draw one particular hand? Solution: In Chapter 13, we found that there are C(52, 5) = 2,598,960 different ways to choose 5 cards from a deck of 52. Each hand is equally likely (has the same chance of being drawn), so the probability of any one particular hand is

13. Equally Likely Outcomes • Experiment: Toss 4 coins • E = { x : x is get 2 heads } • P(E) = ______ ? • Experiment: Roll 2 dice • E = { x : x is sum less than 4 } • P(E) = ______ ?

14. Properties of Probability

15. Counting and Probability Examples • What is the probability in a family with three children that two of the children are girls? Solution: We saw earlier that there are eight outcomes in this sample space. We denote the event that two of the children are girls by the set G = { bgg, gbg, ggb }. • If we draw a 5-card hand from a standard 52-card deck, what is the probability that all 5 cards are hearts? Solution: We know that there are C(52, 5) ways to select a 5-card hand from a 52-card deck. If we want to draw only hearts, then we are selecting 5 hearts from the 13 available, which can be done in C(13, 5) ways.

16. Counting and Probability Examples cont. (2) • Four friends belong to a 10-member club. Two members of the club will be chosen to attend a conference. What is the probability that two of the four friends will be selected? Solution: We can choose 2 of the 10 members in Event E, choosing 2 of the 4 friends, can be done in C(4, 2) = 6 ways. So • What is the probability of getting a 7 when rolling a (fair) pair of dice? • What is the probability of getting 2 Aces when being dealt 2 cards from a standard deck?

17. Probability and Genetics • Y– produces yellow seeds (dominant gene) • g– produces green seeds (recessive gene) • Crossing two first generation plants: Punnett Square

18. Probability and Genetics cont. (2) • Sickle-cell anemia is a serious inherited disease. A person with two sickle-cell genes will have the disease, but a person with only one sickle-cell gene will be a carrier of the disease. If two parents who are carriers of sickle-cell anemia have a child, what is the probability of each of the following: • a) The child has sickle-cell anemia? • b) The child is a carrier? • c) The child is disease free?

19. Odds If a family has 3 children, what are the odds against all 3 children being of the same gender? 6:2 or 3:1 What are the odds in favor? 1:3

20. Odds cont. (2) • A roulette wheel has 38 equal-size compartments. Thirty-six of the compartments are numbered 1 to 36 with half of them colored red and the other half black. The remaining 2 compartments are green and numbered 0 and 00. A small ball is placed on the spinning wheel and when the wheel stops, the ball rests in one of the compartments. What are the odds against the ball landing on red?