[Math] Lottery Ticket Question

data analysispermutationsprobability

A lottery ticket contains six different numbers, chosen from 1 to 39. The winning ticket will match all six numbers in the correct order, plus a bonus number, which may match the other six numbers. The second prize matches the six winning numbers in the correct order, but not the bonus number. What is the probability of winning first or second prize?

Best Answer

The portion "plus a bonus number, which may match the other six numbers. The second prize matches the six winning numbers in the correct order, but not the bonus number", clearly suggests that the bonus number is separately drawn from the $39$ numbers.

Thus there are two cases:

  • $\underline{Bonus\; number\; is\; among\; the\; six\; on\; the\; ticket}$

You can win only the first prize, with $Pr = \dfrac{1}{39\cdot38\cdot37\cdot36\cdot35\cdot34}$

  • $\underline{Bonus\; number\; is\; outside\; the\; six\; on\; the\; ticket}$

You can win the second prize, with $Pr = \dfrac{38}{39}\times\dfrac{1}{39\cdot38\cdot37\cdot36\cdot35\cdot34}$

You can win the first prize, with $Pr = \dfrac{1}{39}\times\dfrac{1}{39\cdot38\cdot37\cdot36\cdot35\cdot34}$

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