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How do i do this Probability question?
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.
1 Answer
- 7 years ago
Let p = P(the number picked is one of the original numbers selected)
Then, p = 2250/150000
Let q = 1-p
(a) Then,
the probability that the jackpot prize is won at any draw
= p + qp + q^2p + q^3p + q^4p +...
= p(1 + q + q^2 + q^3 + q^4 +...)
= p[1/(1-q)] = p/(1-q) (see source)
(b) The probability that the jackpot prize will exceed $250,000 when it is finally won
= P(it more than 25 draws)
= 1 - P(it takes 25 or fewer draws)
P(it takes 25 or fewer draws)
= p + qp + q^2p + q^3p + q^4p +... + q^24p
= p(1 + q + q^2 + q^3 + q^4 +... + q^24)
=p[(1 - q^25) / (1 - q)] (see source)
