Section 7A: Fundamentals of Probability

1. Section 7A: Fundamentals of Probability • Section Objectives • Define outcomes and event • Construct a probability distribution • Define subjective and empirical probabilities • Compute theoretical probabilities • Use the multiplication principle • Compute the probability of an event not occurring Define outcomes and event The set of outcomes is the most basic possible result of an observation or an experiment. For example, if we were to toss a quarter and a dime, we might get a head on the quarter and a tail on the dime (H, T) or we might get a tail on the quarter and a head on the dime (T, H). These are two different and distinct outcomes. An event consists of one or more outcomes that share a property of interest. For example we might be interested in the event that we get one head and one tail. Both of the outcomes above would combine to form that one event.

2. H H H T T H T T first coin second coin Section 7A: Fundamentals of Probability Consider the following experiment: toss two coins sample space

3. Section 7A: Fundamentals of Probability Consider the following experiment: toss two coins 2 H H H T T H T T

4. Section 7A: Fundamentals of Probability Consider the following experiment: toss two coins 2 H H H T 1 1 T H T T

5. Section 7A: Fundamentals of Probability Consider the following experiment: toss two coins 2 H H H T 1 1 T H 0 T T

6. H H H T T H T T Section 7A: Fundamentals of Probability Consider the following experiment: toss two coins 2 1 1 0 Here the event “toss two heads” consists of one outcome {H H}, while the event “toss one head” consists of two outcomes {H T, T H}. Also “toss no heads” consists of the single outcome {T T}. In a more complicated situation (playing cards), the event “full house” consists of many different outcomes: {4♥, 4♣, 4♠, Q♦, Q♥}, etc. How likely is it to draw a full house from five dealt cards? Leads us to probability.

7. Section 7A: Fundamentals of Probability The Probability of an Event is always between P(event A) = 0 P(event A) = 1 zero and 1 (inclusive) means the event is impossible means the event is certain When tossing a fair coin (or coins), we believe that all outcomes are equally likely. This allows us to calculate theoretical probabilities.

8. 2 H H 1 H T 1 T H 0 T T 0 1/4 1 1/2 2 1/4 Section 7A: Fundamentals of Probability Tossing two coins: there are four possible outcomes 1/4 P(2 heads) = 1 in four ways = P(1 head) = 2 in four ways = P(0 or more heads) = P(0 heads) = 1 in four ways = P(3 heads ) = 1/2 1 1/4 0 We now gather the possible values for the random variablex (number of heads) and build the probability distribution: 1

9. Section 7A: Fundamentals of Probability Consider this Experiment: Have three children What are the probabilities associated with the numbers of girls and boys? Build the probability distribution for the random variable x, the number of boys. 0 3 BBB 1/8 BBG 3/8 1 2 3/8 BGB 2 2 1/8 BGG 3 1 GBB 2 GBG 1 GGB 1 GGG 0 NOTE: There are 8 outcomes in the sample space

10. Section 7A: Fundamentals of Probability Define subjective and empirical probabilities Subjective Probabilities are determined by someone making a subjective estimate using experience or intuition. For example, we might want to know the probability that Clemson will beat its arch football rival in the upcoming season: Some will say that we will win, based upon who beat whom when. Some will say that we won’t win, also based upon who beat whom when. This is not very scientific, so we don’t use this method in Mathematics.

11. Section 7A: Fundamentals of Probability Define subjective and empirical probabilities Empirical Probabilities are based upon historical records, observations or experiments. For example, it has rained on June 9th for 30 of the past 40 years, so the empirical probability that it will rain on June 9, 2012, would be calculated as 30/40. Geological records indicate that a river has crested above a particular high flood level just 5 times in the past 1250 years. What is the empirical probability that the river will crest above this flood level next year? Based on the observations of 5 floods in 1250 years, the empirical probability is P(flood) = = .004 5 1250 That is, a 0.4% chance of a flood in any given year.

12. Section 7A: Fundamentals of Probability Compute theoretical probabilities Theoretical probabilities are used when computing the probability of things occurring with equal likelihood. The die could fall on any of six sides, the coin could fall with equal likelihood on heads or tails, etc. Step 1: Count the total number of possible outcomes. Step 2: Among all the possible outcomes, count the number of ways the event of interest, event A, can occur. Step 3: Determine the probability: P(A) = number of ways A can occur total number of outcomes

13. Section 7A: Fundamentals of Probability Example: Assuming that phone numbers are randomly assigned and are equally likely to have any of the ten possible numbers (0-9) in the last position, what is the probability of meeting someone with a phone number that ends in an odd number greater than 2? P(A) = number of ways A can occur P(ends in odd number > 2) = = 10 total number of outcomes number of ways A can occur total number of outcomes

14. Section 7A: Fundamentals of Probability Example: Assuming that phone numbers are randomly assigned and are equally likely to have any of the ten possible numbers (0-9) in the last position, what is the probability of meeting someone with a phone number that ends in an odd number greater than 2? P(A) = number of ways A can occur 4 P(ends in odd number > 2) = = = .4 10 total number of outcomes number of ways A can occur total number of outcomes

15. Section 7A: Fundamentals of Probability Example: Assuming that there are 365 days in the year and that the likelihood of being born on any particular day in the year is the same, what is the probability of meeting someone who was born in either July or August? # days in July = ; # days in August = therefore 31 31 P(A) = 31 + 31 P(July or August) = = = .1699 365 number of ways A can occur total number of outcomes 62 365

16. Section 7A: Fundamentals of Probability Use the multiplication principle The Multiplication Principle is very important for counting outcomes. 2 x 2 x 2 x 2 = 16 outcomes If one process has M outcomes and another process has N outcomes, the total number of possible outcomes for the two processes combined is . M x N If a third process has R possible outcomes and this process is combined with the first two, the number of possible outcomes for the 3-stage process is . M x N x R Example: If you go to a restaurant and there are 6 choices of appetizer, 4 choices of salad, 12 choices of entrée and 7 choices of dessert, how many possible 4-course meals could you order? Solution: The number of possible outcomes is 6 x 4 x 12 x 7 = 2016 meals.

17. Section 7A: Fundamentals of Probability Use the multiplication principle The Multiplication Principle is very important for counting outcomes. 2 x 2 x 2 x 2 = 16 outcomes If one process has M outcomes and another process has N outcomes, the total number of possible outcomes for the two processes combined is . M x N If a third process has R possible outcomes and this process is combined with the first two, the number of possible outcomes for the 3-stage process is x R . M x N Example: If there are three routes between City A and City B and four routes between City B and City C, how many possible routes are there from City A to City C (via city B)? Solution: The number of possible routes is 3 x 4 = 12 routes.

18. Section 7A: Fundamentals of Probability Compute the probability of an event not occurring The probability of event A is P(A). The probability that event A does not occur = 1 – P(A) Example: If the probability that you can win at a particular game is 1/5, the probability that you will lose is 1 – 1/5 or 4/5. Example: If you throw two dice, the probability that you will get a total of 7 is 1/6. The probability that you will not get a total of 7 is . 5/6 the sample space 6 1 P(total of 7) = = 36 6

19. Section 7A: Fundamentals of Probability Compute the probability of an event not occurring The probability of event A is P(A). The probability that event A does not occur = 1 – P(A) Example: If you throw two dice, the probability that you will get a total of less than 9 is 26/36. The probability that you will get a total greater than or equal to 9 is . 10/36 = 1 – (26/36) the sample space 26 P(total < 9) = 36

20. Section 7A: Fundamentals of Probability Suppose we want to know the probability that a 2 appears on a roll of two dice. We can begin by showing the Sample Space. There are 36 outcomes. the sample space If we count the number of outcomes in which a 2 appears, we have such outcomes. The probability of rolling two dice and seeing at least one 2 is . The probability of rolling two dice and NOT seeing a 2 is 11 11/36 25 11 1 – = .6944 P(no 2) = 1 – P(2) = = 36 36

21. Section 7A: Fundamentals of Probability In rolling two dice what is the probability that we will not see a one or a two? Number of outcomes containing a 1 or a 2: 20 the sample space 5 4 5 P(1 or 2) = = 9 9 9 20 1 – = P(No 1 or 2) = 36