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Problem: What's the probability of winning the jackpot of the California lottery? Note, to win the jackpot you must pick 5 numbers from 1-47, and then pick a MEGA number from 1-27. All 5 numbers must match the numbers chosen, and the MEGA number must match the MEGA number chosen.

Solution:

    First, the easy part:

    (a) The probability that you match the MEGA number is simply 1 in 27. The way the game is set up, the MEGA number is completely separate from the other 5 numbers.

    (b) The probability that you match the other 5 numbers is

      1 / (47 choose 5)

      = 1 / (47 * 46 * 45 * 44 * 43) / (5 * 4 * 3 * 2 * 1)

      = 1 / 1533939

    The probability that you match the 5 numbers and the MEGA number, thus winning the jackpot, is

      (1 / 27 ) * (1 / 1533939)

      = 1 / 41,416,353

    Note: the game previously was that you had to choose 6 numbers from 1 to 51 (with no MEGA number). That game was easier, providing you 1 in 18,009,460 odds. (left as an exercise to the reader)

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