Chapter 7

1. Chapter 7 How to Measure Uncertainty With Probability

2. Homework 13 • Read pages pages 454 - 470. • LDI: 7.20, 7.22 7.24 • Exercises: 7.47, 7.48, 7.50, 7.59

3. Random Variables • A random variable is an uncertain numerical quantity whose value depends on the outcome of a random experiment. We can think of a random variable as a rule that assigns one and only one numerical value to each point of the sample space for a random experiment.

4. Example • If you play craps, the sum of the pips on the dice are the random variable. The random experiment is the rolling of the dice.

5. Random Process = tossing a fair coin three times. There are eight possible individual outcomes in this sample space. Since the coin is assumed to be fair, the eight outcomes can be assumed to be equally likely (that is, the probability assigned to each individual outcome is 1/8).

6. 7.4.2 Rules of Probabilities • To any event A, we assign a number P(A) called the probability of the event A. • Assign a probability to each individual outcome, each being a number between 0 and 1, such that the sum of these individual probabilities is equal to 1, and, • The probability of any event is the sum of the probabilities of the outcomes that make up that event. • If the outcomes in the sample space are equally likely to occur, the probability of an event A is simply the proportion of outcomes in the sample space that make up the event A.

7. Discrete Random Variable • A discrete random variable can assume at most a finite or infinite but countable number of distinct values. • Example: Number of eggs a chicken lays in a day. • Example: Number of bombs dropped on a city. • Example: Number of casualties in a battle.

8. Continuous Random Variable • A continuous random variable can assume any value in an interval or collection of intervals. It always has an infinite number of possible values. • Example: Gallons of milk from a cow over its life • Example: Number of hours that CNN broadcasted the Iraq war with out interruption. • Example: Number of hours a battery will run a flashlight.

9. Give the values the random variable can take on: • X is the difference between the number of heads and number of tails obtained when a fair coin is tossed 3 times. • Y is the product of the pips for the roll of 2 fair dice. • R is the time in minutes that this class lasts.

10. A discrete random variable can assume at most a finite or infinite but countable number of distinct values. • A continuous random variable can assume any value in an interval or collection of intervals.

12. Probability Distribution of a Discrete RV • The probability distribution of a discrete random variable X is a table or rule that assigns a probability to each of the possible values of the discrete random variable X.

13. Let X represent the number of people in an apartment. Assume the maximum in a single apartment is 7.

14. Part (b)

15. Let’s Do It • LDI 7.20, page 458

16. The Mean of a Discrete Distribution • The mean of a probability distribution is also called the expected value of the distribution.

17. The Variance and Standard Deviation of a Discrete Distribution • The variance of a discrete probability distribution is • The standard deviation is given by:

18. Good News! • The TI-83 knows how to do these calculations. You simply enter the values of the random variable in L1 and the probabilities in L2 and do the following command: • 1-Var Stats L1,L2

19. Apartments Revisited • What is the expected value, the variance, and the standard deviation of the number of people per apartment?

20. Let’s Do It • LDI 7.22

22. Homework 14 • LDI: 7.26, 7.27, 7.31 • Exercises: 7.60, 7.61, 7.62, 7.65, 7.102, 7.103, 7.104

23. Combinations • “nCr” represents the number of ways of selecting r items (without replacement) from a set of n distinct items where order of selection is not important.

24. Bernoulli Variable • If a random variable has exactly two possible outcome, success and failure and the probability of success remains fixed if the experiment is repeated under identical conditions, then the RV is dichotomous or Bernoulli.

25. Let’s Do It • LDI 7.26

26. Binomial Distribution • A binomialrandom variable X is the total number of successes in n independent Bernoulli trials, on which each trial, the probability of success is p. We say X is B(n,p). • Page 467: Blue box

27. The Binomial Probability Distribution Where p = the probability of success in a single trial q = 1 – p (probability of failure) n = number of independent trials x = number of successes in the n trials

28. Let’s Do It • Page 470: LDI 7.27

29. Mean and Standard Deviation of a Binomial RV

30. Example • A wart remover states it works on 95% of warts. If a total of 10 subjects are selected, what is the probability that 9 of the subjects will have their warts removed?

31. Continuous Random Variables • The probability distribution of a continuous variable X is a curve such that the area under the curve over an interval is equal to the probability that the random variable X is in the interval. The values of a continuous probability distribution must be at least 0 and the total area under the curve must be 1. The uniform and normal distributions we studied in chapter 6 were continuous.

32. Approximating a Discrete RV with a Continuous One • We can use the normal distribution to approximate the binomial when np ≥ 5 and np ≥ 5. • If X is B(n, p) and np ≥ 5 and nq ≥ 5 then X can be approximated by

33. Example • A wart remover states it works on 95% of warts. If a total of 1000 subjects are selected, what is the probability that 900 of the subjects will have their warts removed?