MHA610 Week 2 Discussion Example: Powerball POWERBALL - METHODOLOGY Powerball is a combined large jackpot game and a cash game. Every Wednesday and Saturday night at 10:59 p.m. Eastern Time, we draw five white balls out of a drum with 59 balls and one red ball out of a drum with 35 red balls Source: http://www.powerball.com/powerball/pb_howtoplay.asp As a player, you purchase a ticket on which you specify the numbers [1 through 59] of the five white balls to be chosen, as well as the number [1 through 35] of the one red ball to be chosen. Here's your payoff, again taken from the Powerball website: POWERBALL - PRIZES AND ODDS Match Prize Odds + $10,000 1 in 648,975.96 1 in 19,087.53 $100 1 in 12,244.83 1 in 360.14 $7 1 in 706.43 $4 1 in 110.81 $4 + 1 in 5,153,632.65 $7 + 1 in 175,223,510.00 $100 + Grand Prize $1,000,000 + 1 in 55.41 The overall odds of winning a prize are 1 in 31.85. The odds presented here are based on a $2 play (rounded to two decimal Source: http://www.powerball.com/powerball/pb_prizes.asp MHA610 Week 2 Discussion Example: Powerball We will determine winning and losing probabilities from first principles. First, let's reformat this table: Number of white balls matched 5 4 3 2 1 0 Number of red 1 balls matched 0 You will receive a payoff unless you match 0 red balls and 0, 1, or 2 white balls. [Match 3, 4, or 5 white balls, or the red ball, and you win!] KEY FACTS: We're assuming the lottery is fair: all balls are equally likely to be drawn (classical approach to probability). Also, the selection of the white balls is independent of the selection of the red ball. So we can use the multiplication rule when calculating probabilities. For the red ball: The probability of correctly selecting the red ball from a pool of 35 equally likely possibilities is 1/35. The probability of incorrectly selecting the red ball is 1 - 1/35 = 34/35. For the white balls: 5 matches o There are 59C5 = 5006386 different combinations of 5 numbers drawn from 59, each of which is equally likely (key!). o Only one of these combinations will match the selected 5 numbers. o So the probability of matching all 5 white balls is 1/5006386. MHA610 Week 2 Discussion Example: Powerball 4 matches o There are 5C4 = 5 equally likely ways of choosing 4 winning numbers out of 5, and 59-5 = 54 ways of choosing one non-winning number. [54 = 54C1.] So there are 5*54 = 270 combinations which yield 4 out of 5 winning white balls. 3 matches o There are 5C3 = 10 ways of choosing 3 winning numbers out of 5, and 54C2 = 1431 ways of choosing two non-winning numbers. So there are 10 *1431 = 14310 combinations which yield 3 out of 5 winning numbers. 2 matches o 5C2 * 54C3 = 10 * 24804 = 248040 1 match o 5C1 * 54C4 = 5 * 316251 = 1581255 0 matches o 5C0 * 54C5 = 3162510 Check: 1+270+14310+248040+1581255+3162510 = 5006386 So let's fill in the table with the row and column probabilities: Number of white balls matched 5 4 3 2 1 0 probability Number of red 1 balls matched 0 probability 1/5006386 270/5006386 14310/5006386 248040/5006386 1581255/5006386 3162510/5006386 1/35 34/35 Let T = 35 * 5006386 = 175,223,510. This is the total number of possible combinations of 5 white balls and one red ball, each combination being equally likely. Then, from the multiplication rule, your probability of winning the Powerball lottery with your ticket is: MHA610 Week 2 Discussion Example: Powerball Number of white balls matched 5 4 3 2 1 0 Number of red 1 1/T 270/T 14310/T 248040/T 1581255/T 3162510/T balls matched 0 34/T 9180/T 486540/T 8433360/T 53762670/T 107525340/T Checks: 1+270+14310+248040+1581255+3162510 = 5006386 34+9180+486540+8433360+53762670+107525340 = 170217124 = 34*5006386 5006386 + 170217124 = 175223510 = T Compare with Powerball table: 1/T = 1 in 175223510 = 5.71x10-9 34/T = 1/5153632.65 = 1.94x10-7 270/T = 1/648976 = 1.54x10-6 9180/T = 1/19087.53 = 5.24x10-5 14310/T = 1/12244.83 = 8.167x10-5 486540/T = 1/360.14 = .002777 248040/T = 1/706.43 = .001416 8433360/T = 1/20.777 = .048129 1581255/T = 1/110.813 = .009024 53762670/T = 1/3.259 = .3068 3162510/T = 1/55.406 = .018048 107525340/T = 1/1.63 = .6136 What is the likelihood that you choose a losing ticket, that is, 0 red ball match and 0, 1, or 2 white ball matches? You can sum the specified probabilities, or take (8433360 + 53762670 + 107525340)/T = 169721730/T = .9686. So you have only a 3.14% chance of picking a winning ticket! Powerall: 1/31.85 = .0314