1. Unit 7A Fundamentals of Probability

2. BASIC TERMINOLOGY IN PROBABILITY Outcomes are the most basic possible results of observations or experiments. An event consists of one or more outcomes that share a property of interest. The probability of an event, expressed as P(event), is always a number between 0 and 1 (inclusive). Many times upper case letters are used to represent events. For example, we could say that A is the event of getting “heads” when a coin is tossed. So, P(A) would be the probability of getting “heads” when a coin is tossed.

3. PROBABILITY LIMITS The probability of an impossible event is 0. The probability of an even that is certain to occur is 1. Any event A has a probability between 0 and 1, inclusive.

4. THE MULTIPLICATION PRINCIPLE For a sequence of two events in which the first event can occur M ways and the second event can occur N ways, the events together can occur a total of M × N ways. This generalizes to more than two events.

5. EXAMPLES 1.How many two letter “words” can be formed if the first letter is one of the vowels a, e, i, o, u and the second letter is a consonant? 2.OVER FIFTY TYPES OF PIZZA! says the sign as you drive up. Inside you discover only the choices “onions, peppers, mushrooms, sausage, anchovies, and meatballs.” Did the advertisement lie? 3.Janet has five different books that she wishes to arrange on her desk. How many different arrangements are possible? 4.Suppose Janet only wants to arrange three of her five books on her desk. How many ways can she do that?

6. THREE TYPES OF PROBABILITIES Theoretical Probabilities Empirical Probabilities Subjective Probabilities There are three different types of probabilities.

7. THEORETICAL PROBABILITY A theoretical probability is based on a model in which all outcomes are equally likely. It is determined by dividing the number of ways an event can occur by the total number of possible outcomes. That is,

8. EXAMPLE Find the probability of rolling a “7” when a pair of fair dice are tossed.

9. EMPIRICAL PROBABILITY An empirical probability is based on observations or experiments. It is the relative frequency of the event of interest. In baseball, batting averages are empirical probabilities.

10. COMPUTING AN EMPIRICAL PROBABILITY Conduct (or observe) a procedure a large number of times, and count the number of times that event A actually occurs. Based on these actual results P(A) is estimated as follows:

11. EXAMPLE A fair die was tossed 563 times. The number “4” occurred 96 times. If you toss a fair die, what do you estimate is the probability is for tossing a “4”?

12. SUBJECTIVE PROBABILITY A subjective probability is an estimate based on experience or intuition. EXAMPLE: An economist was asked “What is the probability that the economy will fall into recession next year?” The economist said the probability was about 15%.

13. PROBABILITY OF AN EVENT NOT OCCURRING Suppose the probability of an event A is P(A). Then the probability that the event A does not occur is 1 − P(A).

14. PROBABILITY DISTRIBUTION A probability distribution represents the probabilities of all possible events. It is sometimes put in the form of a table in which one column lists each event and the other column lists each probability. The sum of all the probabilities must be 1.

15. MAKING A PROBABILITY DISTRIBUTION To make a probability distribution: Step 1: List all possible outcomes. Step 2: Identify the outcomes that represent the same event. Find the probability of each event using the theoretical method. Step 3: Make a table in which the first column lists each event and the second column lists the probability of the event.

16. EXAMPLES 1.Suppose you toss a coin three times. Let x be the total number of heads. Make a table for the probability distribution of x. 2.Suppose you throw a pair of dice. Let x be the sum of the numbers on the dice. Make a table for the probability distribution of x.

17. EVENTPROB. 21/36 ≈ 0.028 32/36 ≈ 0.056 43/36 ≈ 0.083 54/36 ≈ 0.111 65/36 ≈ 0.139 76/36 ≈ 0.167 85/36 ≈ 0.139 94/36 ≈ 0.111 103/36 ≈ 0.083 112/36 ≈ 0.056 121/36 ≈ 0.028 Total36/36 = 1.000

18. ODDS The odds for (or odds in favor of) an event A are The odds against (or odds on) an event A are

19. ODDS IN GAMBLING In gambling, the “odds on” usually expresses how much you can gain with a win for each dollar you bet. EXAMPLE: At a horse race the odds on the horse Median are given as 5 to 2. If you bet $12 on Median and he wins, how much will you gain?