November 11, Chapter 8: Probability: The Mathematics of Chance


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1 Chapter 8: Probability: The Mathematics of Chance November 11, 2013
2 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes
3 Probability Rules Probability Rules Rule 1. The probability P(A) of any event A satisfies 0 P(A) 1. Rule 2. If S is the sample space in a probability model, then P(S) = 1. Rule 3. The complement rule: P(A C ) = 1 P(A). Rule 4. The multiplication rule for independent events: P(A and B) = P(A) P(B). Rule 5. The general addition rule: P(A or B) = P(A) + P(B) P(A and B). Rule 6. The addition rule for disjoint events: P(A or B) = P(A) + P(B).
4 Counting distinct items Counting Ordered Collections of Distinct Items Rule A. Suppose we have a collection of n distinct items. We want to arrange k of these items in order, and the same item can appear more than once in the arrangement. The number of possible arrangements is n n n = n k Rule B. (Permutations) Suppose we have a collection of n distinct items. We want to arrange k of these items in order, and any item can appear no more than once in the arrangement. The number of possible arrangements is n (n 1) (n k + 1)
5 Counting Distinct Items Counting Unordered Collections of Distinct Items Rule C. Suppose that we have a collection of n distinct items. We want to select k of those items with no regard to order, and any item can appear more than once in the collection. The number of possible collections is (n + k 1)! k!(n 1)! Rule D. (Combinations) Suppose that we have a collection of n distinct items. We want to select k of these items with no regard to order, and any item can appear no more than once in the collection. The number of possible selections is n! k!(n k)!
6 In poker, a royal flush is a 5card hand containing an ace, king, queen, jack, and 10, all of the same suit. (a) How many royal flush hands are possible? Answer: 4 (b) What is the number of 5card hands possible from a 52card deck? Answer: ( ) 52 5 (c) What is the probability that 5 cards drawn at random from a 52card deck will yield a royal flush? Answer: 4 ( 52 5 )
7 A computer assigns threecharacter login IDs that may contain the digits 0 to 9 as well as the letters a to z, with repeats allowed. 1 What is the probability that your ID contains no x? Answer: ( ) ( ) ( ) What is the probability that your ID contains no digits? ( ) ( ) ( ) What is the probability that your ID contains exactly 2 x s?
8 What is the probability that two people in this room have the same birthday?
9 What is the probability that two people in this room have the same birthday? Answer: 1 365! (365 14)! = 0.223
10 How many different ways can you seat 10 couples at a circular table such that everyone is sitting next to their spouse? 2(9!) Assuming we alternate men and women how arrangements have one couple not sitting together? 8(9!)
11 This time Continuous Probability Models
12 If I let you choose a number between 0 and 1, what is the probability that you will chose a number between.2 and.4?
13 If I let you choose a number between 0 and 1, what is the probability that you will chose a number between.2 and.4? P(.2 X.4) = 0.2
14 Continuous Probability Models Density Curve A density curve is a curve that is always on or above the horizontal axis and has area exactly 1 underneath it. Continuous Probability Models A continuous probability model is a probability model that assign probabilities as areas under a density curve. The area under the curve and above any interval of values is the probability of an outcome in that interval.
15 Would you rather flip a coin for $1 or roll a dice for $30,000?
16 Mean of Probability Model Mean of a Discrete Probability Model Step 1: Make a table with two rows. The first row needs to list all the possible numerical outcome values in the sample space. Step 2: In the second row of the table, list the respective probabilities of each of the outcome values from the first row of the table. Step 3: Write a third row where each entry is the product of the two items in the same column from the first two rows. Now add up all the values in the third row, and you will get the mean of the discrete probability model.
17 Law of Large Numbers Law of Large Numbers Observe any random phenomenon having numerical outcomes with finite mean µ. According to the law of large numbers, as the phenomenon is repeated a large number of times, the proportion of trials in which an outcome occurs gets closer and closer to the probability of that outcome, and the mean x of the observed values gets closer and closer to µ.
18 Standard Deviation Standard Deviation of a Discrete Probability Model Suppose that the possible outcome x 1, x 2,, x k in a sample space S are numbers, and that p j is the probability of outcome x j. The standard deviation of a discrete probability model with mean µ is denoted by the lowercase Greek letter sigma (σ) and is given by this formula: σ = (x 1 µ) 2 p 1 + (x 2 µ) 2 p ) + + (x k µ) 2 p k
19 If I pay you the amount you roll on a dice, what is the mean and the Standard Deviation?
20 If I pay you the amount you roll on a dice, what is the mean and the Standard Deviation? σ = µ = 1 21 ( ) = 6 6 = 7 2 = ((1 3.5)2 + (2 3.5) 2 + (3 3.5) 2 + (4 3.5) 2 + (5 3.5) 2 +
21 Question 1: Should you buy the extended warranty on a new washing machine? Suppose there are two outcomes an 85 percent probability of needing no repairs, and a 15 percent probability of needing no repairs, and a 15 percent probability of needing a $ 200 repairs, and a 15 percent probability of needing a $200 repair during the warranty period. Based on the mean outcome for this model, what would be a breakeven price to you for the extend warranty? Question 2: If you have to pay $ 1 to play a hand of black jack, how much money would you have to win to make it worth playing?
22 An American roulette wheel has 38 slots numbered 0,00, and 1 to 36. The ball is equally likely to come to rest in any of these slots when the wheel is spun. The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains multiples of 3. Joe places a $1 column bet that pay out $ 3 if any of these numbers comes up. 1 What is the probability model for the outcome of one bet, taking into account the $1 cost of a bet? 2 What are the mean and standard deviation for this model? 3 Joe plays roulette every day for years. What does the law of large numbers tell us about this results?
23 Next Time Central Limit Theorem
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