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PowerBall

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In lottery games, the order in which the numbers are chosen is unimportant. All that matters is that you have the same numbers on your card as were drawn by the lottery officials. Since order doesn't matter, we're going to use a combination. We would use a permutation if order mattered. In general, the number of ways to choose r objects from n when order is not important and we are drawing without replacement is given by:

(n,r)= n!/(r!*(n-r)!)

The probability of winning the big jackpot is given by the total number of ways to win divided by the possible number of outcomes. The number of possible outcomes (the denominator) is computed as follows:

There are C(49,5) = 49!/(5! * (49-5)!) = 1,906,884 ways to pick your five numbers. And there are C(42,1) = 42 ways to pick the powerball (you verify the computation using the formula above). Thus there are 1,906,884 * 42 = 80,089,128 total number of ways that the drawing can occur.

To figure out how many ways you can win (the answer is obviously one, but we go through the formalities because we'll need them to do the next part), think of the balls as being partitioned as follows:

Among the 49 numbers, 5 are "winners" and 44 are "losers." Similarly, among the 42 powerball numbers, 1 is a "winner" and 41 are "losers."

If you want to pick all five numbers correctly and pick the powerball correctly, then what you want to do is pick 5 out of the 5 winners and 0 out of the 44 losers on the non-powerball side, and you want to pick 1 out of 1 winners and 0 out of 41 losers on the powerball side. This makes the numerator: C(5,5) * C(44,0) * C(1,1) * C(41,0) = 1 * 1 * 1 * 1 = 1 Hence the probablity is 1/80,089,128

Now let's look at the odds of the "Match 5." Here you have to get all five numbers right on the non-powerball side but the number on the powerball is not important. Using the tools we developed above, you need to pick 5 of the 5 winners on the non-powerball side and 0 of the 44 losers. But this time, we pick 0 of the 1 winners on the powerball

side and 1 of the 41 losers.

Hence the numerator is given by:C(5,5) * C(44,0) * C(1,0) * C(41,1) = 1 * 1 * 1 * 41 = 41

The denominator remains the same for all of these problems. The number of possible powerball combinations never changes. Thus the probability of winning a match 5 is: 41/80,089,128, which is pretty close to 1/1,953,393.

For the "Match 4 + Powerball" the numerator would be:

C(5,4) * C(44,1) * C(1,1) * C(41,0) = 5 * 44 * 1 * 1 = 220

Thus the odds here are 220/80,089,128, which is pretty close to 1/364,042.

You should be able to establish the rest of the formulas pretty easily just by following the pattern that has been established. If you have any questions, please write back.

The overall odds of winning (1:35) can be verified by adding up all of the individual probabilities of winning:

1/74 + 1/118 + 1/605 + ... + 1/80,089,128 = .028706, which is very close to 1/35.

Raining cash and dogs scratch ticket

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The general odds from the National Weather Service Lightning Safety site ()

|Odds of Becoming a Lightning Victim |

|U.S. 2000 Census population as of 2008 |300,000,000 |

|Number of deaths actually reported: 60 |Number of injuries reported: 340 |(total) 400 |

|Estimated number of actual U.S. deaths: 60 |Estimated number of actual injuries: 540 |(total) 600 |

|Odds of being struck by lightning in a given year (reported deaths + injuries) |1/750,000 |

|Odds of being struck by lightning in a given year (estimated total deaths + injuries) |1/500,000 |

|Odds of being struck in your lifetime (Est. 80 years) |1/6250 |

|Odds you will be affected by someone being struck (Ten people affected for every one struck) |1/625 |

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