PLEASE SHOW WORK 16. Probability and Lotteries The Powerball lottery is open to

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PLEASE SHOW WORK

16. Probability and Lotteries The Powerball lottery is open to participants across several states. When entering the powerball lottery, a participant selects five numbers from 1-59 and then selects a powerball number from the digits 1-35. In addition, there's a $1 million payoff for anybody selecting the first five numbers correctly » Show that the odds of winning the Powerball Jackpot are 1 in 175, 223, 510. Show that the odds of winning the $1 million are 1 in 5, 153, 632 On February 18, 2006 the Jackpot reached $365 million. Assuming that you will either win the Jackpot or the $1 million prize, whatâA2s your expected value of winning? Mega Millions is a similar lottery where you pick 5 balls out of 56 and a powerball from 46. Show that the odds of winning mega millions are higher than the Powerball lottery On March 30, 2012 the Jackpot reached $656 million. Is your expected value higher or lower than that calculated for the Powerball lottery?

Explanation / Answer

The Powerball jackpot is won when the 5 numbers are correctly selected from 59 numbers and powerball number is correctly selected from 35 numbers.

Number of ways to choose 5 numbers from 59 numbers = 59C5 = 59! / (5! (59-5)!) = 59! / (5! * 54!) = 5006386

So, the probability to won 5 correct numbers from 59 numbers = 1 / 5006386

The probability to correctly select a powerball number = 1/35

The probability to win Powerball Jackpot = (1 / 5006386) * (1/35) = 1 / 175223510

So, the odds of winning Powerball Jackpot is 1 in 175223510

The $1 million payoff is won when the 5 numbers are correctly selected from 59 numbers and powerball number is not correctly selected from 35 numbers.

The probability to not correctly select a powerball number = 1 - (1/35) = 34/35

The probability to win Powerball Jackpot = (1 / 5006386) * (34/35) = 1 / 5153632

So, the odds of winning $1 million payoff is 1 in 5153632

Given that you will either win the Jackpot or the $1 million prize,

the probability to win Powerball Jackpot = (1 / 175223510) / [(1 / 175223510) + (1 / 5153632)] = 0.0286

and the  probability of winning $1 million payoff = (1 / 5153632) / [(1 / 175223510) + (1 / 5153632)] = 0.9714

Expected value of winning = 0.0286 * $365 million + 0.9714 * $1 million

= $11.4104 million

The Mega millions jackpot is won when the 5 numbers are correctly selected from 56 numbers and powerball number is correctly selected from 46 numbers.

Number of ways to choose 5 numbers from 56 numbers = 56C5 = 56! / (5! (56-5)!) = 56! / (5! * 51!) = 3819816

So, the probability to won 5 correct numbers from 56 numbers = 1 / 3819816

The probability to correctly select a powerball number = 1/46

The probability to win Mega millions Jackpot = (1 / 3819816) * (1/46) = 1 / 175711536

So, the odds of winning Mega millions Jackpot is 1 in 175711536

So, the odds of winning mega millions are higher than the Powerball lottery.

The $1 million payoff is won when the 5 numbers are correctly selected from 56 numbers and powerball number is not correctly selected from 46 numbers.

The probability to not correctly select a powerball number = 1 - (1/46) = 45/46

The probability to win Powerball Jackpot = (1 / 3819816) * (45/46) = 1 / 3904701

So, the odds of winning $1 million payoff in megamillions is 1 in 3904701

Given that you will either win the Jackpot or the $1 million prize,

the  probability to win Powerball Jackpot = (1 / 175711536) / [(1 / 175711536) + (1 / 3904701)] = 0.0217

and the  probability of winning $1 million payoff = (1 / 3904701) / [(1 / 175711536) + (1 / 3904701)] = 0.9783

Expected value of winning = 0.0217 * $656 million + 0.9783 * $1 million

= $15.2135 million

So, the expected value of mega millions is higher than that calculated for the Powerball lottery.

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