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Can someone please show me the lines of working for questions a and b as i am a bit confused as to how to answer the questions.

In a lottery, 2250 numbers are drawn out at random without replacement from a box containing 150,000 ticket numbers. After all the 2250 numbers are drawn, they are returned to the box and a prize winning number is drawn. If this prize winning number is the same as one of the 2250 previously drawn jackpot numbers, a jackpot prize is awarded. If the jackpot prize is not won (that is, the prize winning number is different from one of the 2250 jackpot numbers), then the prize winning number is returned to the box and another prize winning number is drawn. The selection process is repeated until the jackpot prize is won.

a. Find the probability that the jackpot prize is won at any draw.

b. The Jackpot prize is initially $10000 and it increases by $10000 each time the jackpot prize is not won. Find the probability that the jackpot prize will exceed $250,000 when it is finally won.

Samwise · Answer

On any draw that can award the jackpot, the box contains 2250 potential jackpot-winning numbers out of 150,000, so the probability of awarding the jackpot on that draw is
2250/150,000 = 45/3,000 = 3/200
so the probability no jackpot was awarded on that draw is
(200 - 3)/200 = 197/200

a. The jackpot is won on draw N if
(1) it was not won on an earlier draw: probability (197/200)^(N-1); and
(2) a jackpot-winning ticket is drawn on the Nth try: probability 3/200.

So the probability that the jackpot is won on draw N is
3 * 197^(N-1) / 200^N

b. The jackpot on draw N is N * $10,000,
so it exceeds $250,000 if the number of draws exceeds 25.
We're being asked the probability that the number of draws is greater than 25.

That, of course, is precisely the probability that the first 25 draws do not produce a jackpot winner, so it's
(197/200)^25 = 0.985^25
= about 68.5%

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