11. Probability Sample Spaces and Probability

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1 11. Probability 11.1 Sample Spaces and Probability 1

2 Objectives A. Find the probability of an event. B. Find the empirical probability of an event. 2

3 Theoretical Probabilities 3

4 Example A fair coin is tossed; find the probability of getting heads. 4

5 Solution Even though we have not officially defined the term probability, our intuition tells us the following: 1. When a fair coin is tossed, it can turn up in one of 2 ways. Assuming that the coin will not stand on edge, heads and tails are the only 2 possible outcomes. 2. If the coin is balanced (and this is what we mean by saying the coin is fair ), the 2 outcomes are considered equally likely. 3. The probability of obtaining heads when a fair coin is tossed, denoted by P(H), is 1 out of 2. That is, P(H) =. 5

6 Experiments and Sample Space Activities such as tossing a coin (as in the previous Example), drawing a card from a deck, or rolling a pair of dice are called experiments. The set of all possible outcomes for an experiment is called the sample space for the experiment. 6

7 Theoretical Probabilities These terms are illustrated in Table Experiments and Sample Spaces Table

8 Theoretical Probabilities cont d These terms are illustrated in Table Experiments and Sample Spaces Table

9 Event, Favorable Outcome Returning to the Example( tossing a coin ), we see that the set of all possible outcomes for the experiment is = {H, T}. But there are only two subsets of that can occur, namely, {H} and {T}, and each of these is called an event. If we get heads that is, if the event E = {H} occurs we say that we have a favorable outcome or a success. Since there are 2 equally likely events in is E, we assign the value to the event E. and 1 of these 9

10 Example Suppose that a fair coin is tossed 3 times. Can we find the probability that 3 heads come up? As before, we proceed in three steps as follows: 1. The set of all possible outcomes for this experiment can be found by drawing a tree diagram, as shown in Figure Tree diagram. Figure

11 Example As we can see, the possibilities for the first toss are labeled H and T, and likewise for the other two tosses. The number of outcomes is The 8 outcomes are equally likely. 3. We conclude that the probability of getting 3 heads, denoted by P(HHH), is 1 out of 8; that is, P(HHH) =. If we want to know the probability of getting at least 2 heads, the 4 outcomes HHH, HHT, HTH, and THH are favorable out of the 8 outcomes shown in Figure 11.2, so the probability of getting at least 2 heads is =. 11

12 Definition of the Probability of an Event In examples such as these, in which all the possible outcomes are equally likely, the task of finding the probability of any event E can be simplified by using the following definition: 12

13 Example Ten balls numbered from 1 to 10 are placed in an urn. If 1 ball is selected at random, find the probabilities of the following: (a) An even-numbered ball is selected (event E). (b) Ball number 3 is chosen (event T). (c) Ball number 3 is not chosen (event T ). 13

14 Solution (a) There are 5 outcomes (2, 4, 6, 8, 10) in E out of 10 equally likely outcomes. Hence, P(E) = =. (b) There is only 1 outcome (3) in the event T out of 10 equally likely outcomes. Thus, P(T) =. (c) There are 9 outcomes (all except the 3) in T out of the 10 possible outcomes. Hence, P(T ) =. 14

15 In the previous Example we found P(T) = and P(T ) =, so P(T ) = 1 P(T). This is a general result because T T = and T T =. Thus, Therefore, Remark n(t T ) = n(t) + n(t ) = n( ) or, by the definition of the probability of an event, 15

16 Probability of an Event Not Occurring Thus, the probability P(T ) of an event not occurring is 1 P(T). 16

17 Example A coin is thrown 3 times. Find the probability of obtaining at least 1 head. 17

18 Let E be the event that we obtain at least 1 head. Then E is the event that we obtain 0 heads; that is, that we obtain 3 tails. From the preceding discussion, P(E) = 1 P(E ). Here, P(E ) is the same as P(TTT) = ; hence, P(E) = 1 P(TTT) = 1 =. Solution 18

19 Empirical Probability 19

20 Empirical Probability Because the probabilities in the preceding examples are based on the theory that the outcomes are equally likely (a balanced coin, fair dice), they are called theoretical probabilities. After performing an experiment in which the coin was tossed 250 times and the number of heads observed was 140, we concluded that the probability of heads for this coin was 20

21 Empirical Probability of an Event E P(H) is called the empirical (expected or experimental) probability of heads. In general, 21

22 Example An online survey of 324 people conducted by Insight Express asked the question, What is your primary credit card? The results are shown in the bar graph. 22

23 Example If a person is selected at random from the 324 surveyed, what is the empirical probability that (a) the person s primary card is MasterCard? (b) the person s primary card is Visa? (c) Which event has the highest empirical probability? What is that probability? (d) Which event has the lowest empirical probability? What is that probability? (e) If you were the manager of a retail store and you can only accept two types of credit cards, which two cards would you accept? 23

24 Solution (a) According to the graph, 94 out of 324 people use MasterCard as their primary card; thus, the empirical probability of selecting a person whose primary card is a MasterCard is (b) Similarly, 175 people use Visa as their primary card; thus, 24

25 Solution cont d (c) The event with the highest empirical probability (longest bar in the graph) corresponds to the selection of a person who uses Visa as his or her primary card. The probability is (d) The event with the lowest empirical probability (shortest bar in the graph) corresponds to the selection of a person who uses Diner s Club as his or her primary card. The probability of that is 25

26 Solution (e) Visa and MasterCard (those are the cards that most people in the survey use as their primary card) 26

Coin-Toss Simulator The formulas to find the theoretical or empirical probability of an event are very similar. Is there a relationship between the numerical results obtained when using the formulas?

27 Coin-Toss Simulator The formulas to find the theoretical or empirical probability of an event are very similar. Is there a relationship between the numerical results obtained when using the formulas? Suppose somebody claims to have a fair coin. We can do it by performing an experiment in which we toss the coin 1, 10, 100, 1000, and 10,000 times. But who has the time to do this? We used a coin-toss simulator (see Figure 11.5). Figure

28 28

29 Law of Large Numbers What do you notice about the decimal value of the empirical probabilities (0.7, 0.54, 0.493, , and )? As the number of tosses gets bigger, the value gets near 0.5, the theoretical probability. This is a good indication (though not a proof) that the coin is fair. More importantly, it illustrates the fact that as an experiment is repeated a large number of times, the empirical probability of an event tends to get closer to the theoretical probability of the event. This principle is appropriately known as the Law of Large Numbers. 29

30 Oddly enough, if you stand 100 coins on edge on a table and slam the table causing the coins to topple, more than half of the coins will show heads! As the number of tosses gets bigger, the theoretical probability for heads is still higher than 0.5! 30

Events A and B are said to be independent if the occurrence of A does not affect the probability of B.

Independent Events Events A and B are said to be independent if the occurrence of A does not affect the probability of B. Probability experiment of flipping a coin and rolling a dice. Sample Space: {(H,