Question 1207679: The odds of winning the Powerball Lottery are 1 in 292 million. How are the odds calculated ?
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Answer by math_tutor2020(3817)   (Show Source):
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Powerball consists of selecting 6 numbers.
The first five of which range from 1 to 69 where repeats are not allowed.
The 6th number, known as the Powerball, can be a mirror copy of any of the previous 5.
The list of choices for this 6th ball are from this set {1,2,3,...,25,26}
Here are some example selections
1,2,3,4,5,25
1,2,3,4,5,5
30,23,69,17,32,12
40,48,1,12,46,22
58,13,10,64,6,26
30,10,15,9,3,1
Each row is a different lotto ticket.
As we'll determine in the next section, there are roughly 292 million of these combos.
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When matching up the 5 balls, the order doesn't matter.
Something like {1,2,3,4,5} is the same as {5,4,3,2,1}.
Since order doesn't matter we'll use the nCr combination formula.
n = 69 and r = 5
n C r = (n!)/(r!(n-r)!)
69 C 5 = (69!)/(5!*(69-5)!)
69 C 5 = (69!)/(5!*64!)
69 C 5 = (69*68*67*66*65*64!)/(5!*64!)
69 C 5 = (69*68*67*66*65)/(5!)
69 C 5 = (69*68*67*66*65)/(5*4*3*2*1)
69 C 5 = 1348621560/120
69 C 5 = 11238513
There are exactly 11,238,513 ways to select the five regular balls where repeats aren't allowed, order doesn't matter, and you select in a range from 1 to 69.
Then we multiply this result with 26 since there are 26 values to pick for the Powerball.
26*11,238,513 = 292,201,338
This is roughly 292 million
It represents the total number of possible combinations, i.e. the total number of ways to play the game.
So that's where the "1 in 292 million" comes from.
The 1 refers to the only one way to win the grand prize is to get all 6 matching numbers.
The order of the initial 5 balls doesn't matter.
Side notes:- If a billionaire wanted a guaranteed way to win the grand prize, all they have to do is buy all 292 million lotto combos. Depending on the jackpot size, they will likely lose more money than they gain.
- To confirm the answer, search out "powerball odds". The number 292,201,338 should be found in one of the results.
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