# Problem 70 Solution

The number of ways the six balls can be drawn are (39:5)*34 = (39!/(34!*5!) * 34 = 19,575,738.

Lets assume the chosen numbers are 1-2-3-4-5. Of the 19,575,738 ways to arrange the six drawn balls how many will fit the drawn numbers to make a winner? Take the ratio of that number of 19,575,738 and you have the probability.

There are 34 ways to draw the 6 numbers to match 1-2-3-4-5 (you must match all of the first 5, the bonus ball can be any of the remaining 34).

The number of ways to match four plus the bonus ball are 170 (the bonus ball can be any of the 5 you got right, and the last ball can be any of the remaining 34).

The number of ways to match four w/o the bonus ball are 5610 (5 different ways to arrange the 4 correct numbers * the 5th number can be any of the remaining 34 * the bonus ball can be any of the remaining 33).

Keep following the above logic for all possible winning combinations:

```                 Ways     Probability           Expected
Type of win      to win   of winning   Payoff   return
--------------   ------   -----------  ------   --------
Pick 5               34   .0000017368  \$50000   .0868
Pick 4 + BB         170   .0000086842    \$600   .0052
Pick 4 w/o BB      5610   .0002865792    \$400   .1146
Pick 3 + BB       11220   .0005731585     \$30   .0172
Pick 3 w/o BB    179520   .0091705355     \$15   .1376
Pick 2 + BB      179520   .0091705355      \$2   .0183

Total            376074                         .3798
```

Note: In all fairness to the state I should mention that you can buy three tickets for the price of two, increasing the expected return to 56.97 cents (but still a waste of money).

Michael Shackleford, A.S.A.