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## Lotteries - UTEP MATHEMATICS

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Cash-3, Powerball, Mega Millions, and other ways to waste your money
Lotteries
Types of Lotteries
Scratch-off TicketsReady-printed numbersUser-selected number gamesVideo based games
User-Selected Number Games
One-numberMulti-numberMulti-number with two sets of numbersKeno type
Cash 3
Pick a three digit number from 000 – 999. Say, 123.Two main types of bets:Straight bet indicates you think it will be an exact match. That is, you think the number 123 will win.Box bet indicates that you think it will be those digits, but a possibly different order. You can win with 213, 312, 321, 132, 231 as well as 123.Other bets may include combo, front pair, back pair, and split, but these depend on the state in which the lottery is held.
Cash 3 Questions and Examples
How many numbers are there from 000 – 999?If you choose a straight bet, how many different winning numbers are there?A probability is defined as the number of favorable outcomes (what you want to happen) divided by the total number of possible outcomes. What is the probability you win a Cash 3 straight bet?
We have the same 1000 numbers from 000 – 999. This time we are going to place a box bet. For simplicity, we will use three different digits. Our pick is 456. How many different ways could we win a box bet in a Cash 3 game?What is the probability that you win a box bet in a Cash 3 game?
Straight vs. Box
Which has a greater likelihood of happening: winning a straight bet or a box bet?Which one do you think pays better? Why?
Multi-number with two sets of numbers
PowerballDraw 5 white balls out of a drum with 59 white balls and one red ball out of a drum with 35 red balls.Cost to play: \$2.There are 9 ways to win!5 white + 1 red wins the Jackpot - odds 1 in 177,223,5105 white only wins \$1,000,000 – odds 1 in 5,153,632.654 white +1 red wins \$10,000 – odds 1 in 648,975.96
Still winning….4 white only wins \$100 – odds 1 in 19,087.963 white + 1 red wins \$100 – odds 1 in 12,244.833 white wins \$7 – odds 1 in 360.142 white + 1 red wins \$7 – odds 1 in 706.431 white + 1 red wins \$4 – odds 1 in 110.811 red wins \$4 – odds 1 in 55.41.These odds are extremely difficult to calculate as they take into consideration many outcomes. We will accept the lottery’s numbers for the purposes of this class.
Mega MillionsDraw 5 numbers from 1 to 56 and one number from 1 to 46.Megapliercost extra, but can double, triple or even quadruple your winnings.Cost of ticket: \$1 (plus another \$1 formegaplier)11 ways to win!!!Match all to win the Grand Prize – odds 1 in 175,711,536
Still winning…Match 5 of first set win \$250,000 – odds 1 in 3,904,700.8Match 4 of first set + second set win \$10,000 – odds 1 in 689,064.8471Match only 4 of first set win \$150 – odds 1 in 15,312.5521Match 3 of first set + second set with \$150 – odds 1 in 13,781.2969Match only 3 of first set win\$7 – odds 1 in 306.2510Match 2 of first set + second set win \$10 – odds 1 in 843.7529
Match 1 of first set + second set win \$3 – odds 1 in 140.6255.Match only second set win \$2 – odds 1 in 74.8008.Match two of first set (1 in 18.7501), one of first set (1 in 3.125) or no matches (1 in 1.6622) you win \$0.
Pay off vs. Expected Value
We know we can win the jackpot, or even \$1,000,000 in the lottery and this is why we play. This is the pay off that lures us in.Mathematically speaking, the pay off is misleading.Expected value is a way of calculating how much you can expect to win, or lose, over the long range course of the game.
Expected value formula
If you have three possible outcomes each with probability p1, p2 and p3, and winnings of w1, w2, and w3, respectively, you can find the expected value with the formula:
That is, multiply the probability of an outcome by the winnings (orlosings) for that outcome.Example: Suppose I play a game with dice. If the number rolled is a 1 or 2, I win \$7. If the number rolled is a 3 or 4, I win \$3, and if the number rolled is a 5 or 6 I lose my bet of \$4. Find the expected value.
Another Example
You have 10 marbles in a bag: 4 red, 2 blue, 3 orange and 1 yellow. You pay \$1 to play the game. You lose your bet if you draw a red, you win \$10 if you draw a yellow, you win \$5 if you draw a blue, and you win \$2 if you draw an orange marble. What is the expected value of this game?

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