advertisement

STAT 110 - Section 5 Lecture 23 Professor Hao Wang University of South Carolina Spring 2012 Example on tree diagram There is an 80% chance you will make the first red light driving in to work. If you make the first, there is a 90% chance you will also make the second. If you miss the first, there is a 90% chance you will also miss the second. What is the chance you get stopped by both lights? A) 0.08 = 8% D) 0.18 = 18% B) 0.09 = 9% E) 0.72 = 72% C) 0.16 = 16% Example on sampling distribution A simple random sample of 1000 college students is taken and asked if they agree that “The dining choices on my campus could be better. If the true percent for the whole population is 60%, the percent in the sample who will say “Yes” should be close to a normal distribution with mean 0.6 and standard deviation of about 0.015. What’s the probability that 0.63 or more in the sample will say “Yes”? A 2.5% B 5% C 10% D 60% E 1.5% Chapter 20: Expected Value How much is a Pick 4 lottery ticket worth? In a Pick 4 game, you make a Quick-Pick $1 Straight-Box wager. The computer chooses a four digit number, and you have two ways to win. 1. You win $3,100 if you exactly match the winning number. 2. You win $600 if your number has the same four digits as the winning number, in any order. The probability model according to their web-site is: outcome $0 probability 0.9996 $600 0.0003 $3,100 0.0001 expected value – found by multiplying each outcome by its probability and then summing over all possible outcomes possible outcomes: a1, a2, …, ak probabilities: p1, p2, …, pk expected value = a1 p1 + a2 p2 + … + ak pk outcome $0 probability 0.9996 $600 0.0003 $0 + $0.18 + $3,100 0.0001 $0.31 = $0.49 • Suppose a sorority is selling 1000 raffle tickets for $1.00 each. One ticket will be drawn at random and the winner receives $200.00. There are two possible outcomes here: 1. Win $200.00 (so net gain is $199 with probability 1/1000) 2. Do not win (so net “gain” is -$1 with probability 999/1000) So how would we find the average net gain for $1 ticket? • An insurance company sells a particular policy that costs the insured $1,000 to purchase. The insurance company knows that 1/100 of these policy holders will file a claim worth $20,000, 1/200 will file a $50,000 claim, and 1/500 will file a claim of $100,000. Find the insurance company’s expected net gain for one such policy. You play a game with two possible outcomes. Outcome A has probability 0.4 and outcome B has probability 0.6. When B occurs you win $2.00; otherwise, you lose $1.00. What is your expected value for this game? A. $2.00 B. $0.80 C. $0.20 D. -$0.80 • Suppose we have the following distribution of outcomes for a “loaded” die. • Outcome 1 2 3 4 5 6 • Probability 0.5 0.1 0.1 0.1 0.1 0.1 • Find the expected number you’ll get when rolling this die. A.1 B. 1.5 C. 2 D. 2.5 E. 3 Expected value and house edge Roulette. Say we bet $10 on Black. There are, 0 to 36, so 37 possible outcomes. The possible outcomes are anyone of 18 possible Blacks, or anyone of 18 possible Reds, Or Green (don't forget that with roulette the zero is Green). 1. 2. Doubling money if black (so net gain is $10 ) Losing money otherwise (so net “gain” is -$10) What’s the chance of doubling money ? A 1/37 B 36/37 C 19/37 D 10/37 E 18/37 Roulette. Say we bet $10 on Black. There are, 0 to 36, so 37 possible outcomes. The possible outcomes are anyone of 18 possible Blacks, or anyone of 18 possible Reds, Or Green (don't forget that with roulette the zero is Green). 1. 2. Doubling money if black (so net gain is $10 ) Losing money otherwise (so net “gain” is -$10) What’s expected (average) net gain for a $10 bet ? A $-10/37 B $10/37 C $1/37 D $-1/37 E $18/37 Roulette. Say we bet $10 on Street. There are, 0 to 36, so 37 possible outcomes. 1. 2. Win $120 if any three numbers horizontal (so net gain is $110 ) Lose money otherwise (so net “gain” is -$10) What’s the chance of winning ? A 1/37 B 33/37 C 19/37 D 3/37 E 12/37 Roulette. Say we bet $10 on Street. There are, 0 to 36, so 37 possible outcomes. 1. 2. Win $120 if any three numbers horizontal (so net gain is $110 ) Lose money otherwise (so net “gain” is -$10) What’s expected (average) net gain for a $10 bet ? A $-10/37 B $10/37 C $1/37 D $-1/37 E $18/37 The Law of Large Numbers law of large numbers If a random phenomenon with numerical outcomes is repeated many times independently, the mean of the actually observed outcomes approaches the expected value. • In many independent repetitions, - the proportion of each possible outcome will be close to its probability. - the average outcome obtained will be close to the expected value. The Law of Large Numbers Application • The Law of Large Numbers explains why gambling is a business for a casino – they use the fact that the average winnings of a large number of customers will be quite close to its expected value and that is how they make money • Other areas: Insurance, lottery, etc