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I came across the following problem involving the Powerball lottery.

The Powerball lottery has the following rules. There are $69$ white balls (numbered from $1$ to $69$) and $26$ red balls (numbered from $1$ to $26$). At each drawing five white balls and a red ball are chosen uniformly at random. The result of the drawing is the five white numbers are listed in decreasing order and the red number.

Find the probability that the picked six numbers are all different.


So far I have that the number of ways to pick the first $5$ white balls is equal to $69\cdot68\cdot67\cdot66\cdot65.$ However I am having trouble understanding how to account for the red powerball, which has a different range of possible outcomes, being $1$ to $26.$ So that when if all five white balls are chosen over $27$ for instance, we still have $26$ choices for the red powerball, but if all five white ball are chosen below $26,$ now we have only $21$ choices for the powerball.


Any hints on how to proceed are greatly appreciated.

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This question can be done by decomposing the event into 2 parts: The thing about decreasing order is completely irrelevant to the actual solution. Part 1: Choosing a red ball. There are 26 choices. Part 2: Choosing the 5 white balls. This is $68\choose5$. We set 68 because we want to avoid the ball we already picked( the red one) Multiply them together. Divide by the total amount of choices, which is 26 times $69\choose5$. The result is your probability

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Some hints

  • Consider that a coincidence can happen only if the red ball equals one of the white balls.

  • Compute the probability of coincidence given that (conditioned to) the red ball is $1$

  • Compute the probability of coincidence (total probability)

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