Probability

1. Probability Parts of life are uncertain. Using notions of probability provide a way to deal with the uncertainty.

2. Probability is a numerical measure of the likelihood that an ‘event’ will occur. Probability values range from 0 to 1. 0 means the event will not occur. 1 means the event will surely occur. .5 means that an event is as likely to occur as not occur. The closer the probability is to 1, the more likely the event is to occur. Note in the definition of probability the word event is used. We will come back to the word event, but now we turn to the idea of an experiment. An experiment is a process that generates well defined outcomes, or what we call sample points.

3. Some examples of experiments are: - toss a coin, with sample points being heads, tails. - conduct a sales call, with sample points being sale, no sale. The sample space for an experiment is the set of all experimental outcomes. In order to assign probabilities one has to be able to count all the experimental outcomes. Next let’s study some rules to help us think about how to count all the experimental outcomes.

4. Multiple-Step Experiments If an experiment can be described as a sequence of k steps with n1 possible outcomes on the first step, n2 possible outcomes on the second step, and so on until we get to nk, then the total number of experimental outcomes is the product (n1)(n2)...(nk). Say a construction project has two stages - design and construction. If the design could be completed in 2, 3, or 4 months and the construction could be done in 6, 7, or 8 months, then there are (3)(3) = 9 different experimental outcomes. I will list the outcomes as ordered pairs of numbers, with the first number the time to complete the design and the second the time to complete the construction: (2, 6), (2, 7), (2, 8), (3, 6), (3, 7), (3, 8), (4, 6), (4, 7), and (4, 8).

5. Combinations Another type of experiment consists of taking n objects from a larger set of N objects. An example of this might be a case where there are 5 job applicants, but only 2 jobs. How many different experimental outcomes are there? (by the way some outcomes may have a greater chance of occurring because some of the people may have more favorable characteristics - but here we just want to list possible outcomes.) We will use factorial notation here - this is a math thing. For example, 5! = (5)(4)(3)(2)(1) =120 The formula for the number of combinations when taking n from N is N!/[n!(N-n)!]. For example from above 5! = (5)(4)(3)(2)(1) = (5)(4) = 10 2!(5 - 2)! 2!(3)(2)(1) 2

6. Permutations Remember having a lock at school? The dial on the lock might have had 40 numbers. To open the lock you spun the dial to the right several times and settled on the first number in the combination, then you went around to the left once past that number and then settled on the second number, then you went right to the third number. Say you had the combination 7 - 16 - 32. This is 3 numbers from 40. Is the combination 32 - 7 - 16 the same as 7 - 16 -32? The answer is no. Parker, what is the point? The idea of a combination on the previous screen dealt with combinations where order did not matter. 7 - 16 - 32 and 32 - 7 - 16 would be the same and counted once. But in a permutation they are different. Perhaps a better name for our locks would be permutation locks – order matters.

7. The formula for the number of permutations when taking n from N is N!/(N-n)!. For example 2 from 5 is 5! = (5)(4)(3)(2)(1) = (5)(4) = 20 (5 - 2)! (3)(2)(1) Let’s do another example. Say we have the letters A, B, and C. Say we want to choose 2 of these. We have 3!/(3-2)! = 3(2)/1 = 6. The permutations would be AB, BA, AC, CA, BC, CB, but combinations are AB, AC, BC. So, there are less combinations than permutations.

8. Assigning Probabilities Each experimental outcome must be assigned a probability of 0, or 1, or somewhere between 0 and 1. The other feature of assigning probabilities is that the sum of the probabilities of all the experimental outcomes must be 1. There are three ways of assigning probabilities: the classical method, the relative frequency method and the subjective method. The classical method is used when each of the experimental outcomes is equally likely to occur. If there are t experimental outcomes, then the probability of any one experimental outcome is 1/t. An example is a roll of a die. There are 6 possible outcomes and each one has probability 1/6.

9. The relative frequency method is used when data is available about the past history of the experiment. The probability of an outcome is the relative frequency of the outcome. You can form this probability by taking the ratio of the number of times the outcome came up over the total number of times of the experiment. As an example, if out of 100 sales calls, you had 37 sales, the probability of a sale would be 37/100 = .37. The subjective method is used when the other two can not be used. You and a friend may consider the experiment that Nebraska will win the national championship in football this year. The outcomes are either they will win or they will not win. You may say P(win) = .4 (meaning the probability of a win is .) Thus you also say P(not win) = .6. Your friend might say the probabilities should be .7, .3, respectively. Your differences represent your subjectivity.

10. Let’s do a problem. Say we have a population of 50 bank accounts and we want to take a random sample of four accounts in order to learn about the population. How many different random samples of four accounts are possible? Since we do not have a situation like a ‘permutation’ lock (once account is chosen, its in), we use the combination formula: 50! = (50)(49)...(2)(1) = (50)(49)(48)(47) 4!(46!) 4!(46)(45)...(2)(1) (4)(3)(2)(1) = 230300. Let’s do another example on the next slide.

11. Say you have an ordinary deck of playing cards - you know, ace through king in spades, hearts, diamonds, and clubs. So there are 52 cards in the deck. In many games of poker you get 5 cards. How many different combinations of 5 cards are there? 52! = (52)(51)(50)(49)(48) = 2598960 5!(47!) (5)(4)(3)(2)(1) This means there are 2 million, 598 thousand, 960 different combinations of hands you could be dealt. Most hands you get are not memorable - you know, you get a 7 of hearts, queen of spades, 3 of clubs, 9 of clubs and a 4 of spades. But a royal flush hearts - 10, jack, queen, king, and ace all hearts - is memorable. Each hand mentioned has a 1 divided by 2598960 chance of happening. But the royal flush is a winner!