At a fete, cards bearing numbers 1 to 1000, one number on one card, are put in a box. Each player selected one card at random and that card is not replaced. If the selected card has a perfect square greater than 500, the player wins a prize. What is the probability that

(i) the first player wins a prize?

(ii) the second player wins a prize, if the first has won?

Given that, at a fete, cards bearing numbers 1 to 1000 one number on one card, are put in a box. Each player selects one card at random and that card is not replaced so, the total number of outcomes are n(S) = 1000

If the selected card has a perfect square greater than 500, then player wins a prize.

(i) Let E_{1} = Event first player wins a prize = Player select a card which is a perfect square greater than 500

= {529, 576, 625, 729, 784, 841, 900, 961}

= {(23)^{2},(24)^{2},(25)^{2},(26)^{2},(27)^{2},(28)^{2},(29)^{2},(30)^{2},(31)^{2}}

∴ n(E) = 9

So, required probability

(ii) First, has won i.e., one card is already selected, greater than 500, has a perfect square. Since, repetition is not allowed. So, one card is removed out of 1000 cards. So, number of remaining card is 999.

∴ Total number of remaining outcomes, n(S’) = 999

Let E_{2} be the event that the second player wins a prize, if the first has won.

Then, the remaining cards has a perfect square greater than 500 = 8

So, required probability

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