Powerball 55
Odds.
How to calculate odds? The odds, 1 in N, mean that the probability of winning is approximately 1/N.
We will write nCr as the name for "n choose r", which is the number of ways r objects can be chosen out of n objects. It is the rth item in the nth row of Pascal's triangle.
5 balls can be drawn from 55 balls in (55 nCr 5) ways. (55 nCr 5) = 3,478,761.
Guessing correctly k of them can be done in (5 nCr k)*(50 nCr (5-k)) ways.
(Note that (5 nCr k) is a correct choice, and (50 nCr (5-k)) is an incorrect choice.)
So, you can pick 1 in 1,151,500 ways, 2 in 196,000 ways, 3 in 12,250 ways, 4 in 250 ways, and 5 in 1 way. (Check these on the TI-34/83/84.)
Thus the probability of guessing k balls correctly is
(5 nCr k)*(50 nCr (5-k))/(55 nCr 5)
And because guessing the power ball correctly has probability 1/42, and guessing it incorrectly has probability 41/42, the probabilities are:
k correct with powerball: (5 nCr k)*(50 nCr (5-k))/(55 nCr 5)/42
k correct without powerball: (5 nCr k)*(50 nCr (5-k))/(55 nCr 5)*41/42
The reciprocals of these numbers, rounded to whole numbers, are listed on the Powerball web
page for k = 0, 1, ..., 5, with one exception (k = 0, no powerball). They are also given below:
|
Match: |
Prize: |
Apprx. odds per $1 play: |
|
|
Jackpot* |
1 in 146,107,962 |
|
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$200,000** |
1 in 3,563,609 |
|
|
$10,000** |
1 in 584,432 |
|
|
$100** |
1 in 14,254 |
|
|
$100** |
1 in 11,927 |
|
|
$7** |
1 in 291 |
|
|
$7** |
1 in 745 |
|
|
$4** |
1 in 127 |
|
|
$3** |
1 in 69 |
|
Overall odds of winning a prize are approximately 1 in 36.6 |
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Tasks.
1. Enter the numerical expressions above the table in the TI-34II, and recompute all odds.
One way to do this:
555 nCr 5 → B
OP1=(5 nCr A)*(50 nCr (5-A))/B/42→C gives probability of A correct with powerball
OP2=C*41 gives probability of A correct without powerball
2. Show how the overall odds of winning, 1:36.6, are computed.
1/36.6 = 0.02732...
and
1/69 + 1/127 + 1/745 + ... + 1/146.107,962 ≈ 0.027.
About this "game".
(1) This is not really a game, because the odds are not reasonable enough for any gambler. A person buying 1000 tickets per year expects to win:
jackpot (all 5 plus powerball) - once every 146,108 years.
$200,000 (all 5) - once every 3563 years
$100 (any 4 of 5, or any 3 of 5 + powerball) - once every 12 years
(The reciprocal of the probability divided by the number of tickets bought each year.)
(2) The game is pari-mutuel. This means that if more than one person wins one of the high prizes, the money is divided among the winners, so, for example, instead of $200,000,
each of three winners would get $66,666.66, before taxes. (All gambling wins are taxable.)
(3) The jackpot is not paid to the winner. If the winner wants cash, he/she gets only 50% of the jackpot, and Powerball keeps the rest. Otherwise the winner gets the interest on the jackpot at the rate of 4% per year for 25 years, and the company keeps all the money. (It is called paying the jackpot in 25 installments!)
Why do people play?
This is not a math question, but it can be answered. A lottery ticket is a "license to
dream about riches". Hoping to get rich by finding a hidden treasure may look childish and naive. But hoping to win a lottery is respectable and usually harmless, providing that not too much money is spent on tickets. The small amounts occasionally won are important incentives, they are proofs that winning is possible, and they keep hope alive.