29. A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that (i) there is at least an even chance of drawing a heart. (ii) the probability a greater than 3/4?

Dear Student,
Solution)
(i) Let p denote the probability of drawing a heart from a deck of 52 cards. So,

 p=1352=14and q=1-q=1-14 = 34

Let the card be drawn n times. So, binomial distribution is given by: P(X=r)=Crnprqn-r
Let X denote the number of hearts drawn from a pack of 52 cards.
We have to find the smallest value of n for which P(X=0) is less than 14
P(X=0) < 14
C0n14034n-0<1434n<14Put n=1, 341 not less than 14      n=2, 342 not less than 14       n=3, 343 not less than 14So, smallest value of n=3
Therefore card must be drawn three times.

(ii) Given the probability of drawing a heart > 34
1 - P(X=0) > 34
1- C0n14034n-0>341-34n>341-34>34n14>34n

For n=1, 341 not less than 14 n=2, 342 not less than 14 n=3, 343 not less than 14 n=4, 344 not less than 14 n=5, 345 not less than 14
So, card must be drawn 5 times.
Regards!

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