Question

One thousand cards are drawn with replacement from a standard deck of 52 playing cards, and let X be the total number of aces drawn. Find the approximate probability that 65 ≤ X ≤ 90.

Answer 1

Answer :

Given that :

No.of cards drawn = n = 1,000

Total Playing cards = 52

In cards total no.of ace cards = 4

then,

probability of ace p = 4 / 52 = 0.077

q = 1 - p = 1 - 0.077 = 0.923

Mean = = n * p = 1000 * 0.077 = 77

Standard deviation = = npq = 1000 * 0.077 * 0.923 = 71

Therefore,

probability of P(65 < X < 90) = P(X < 90) - P(X < 65)----------->(1)

Consider,

P(X < 90) is :

= (90 - 77) / 71

= 13 / 71

= 0.18

P(X < 90) = P(Z < 0.18) = 0.5714

Now,Consider,

P(X < 65) is :

= (65 - 77) / 71

= -12 / 71

= -0.17

P(X < 65) = P(Z <-0.17) = 0.4325

Substitute these two values in equatioin 1,

P(65 < X < 90) = P(X < 90) - P(X < 65)

= 0.5714 - 0.4325

= 0.1389

P(65 < X < 90) = 0.1389