Multiple choice

A card is drawn and replaced in an ordinary pack of playing cards. The number of times a card must be drawn so that the probability of getting atleast a club card is greater than $\displaystyle \frac{3}{4}$ is

  1. $7$
  2. $6$
  3. $5$
  4. $4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Probability of not getting a club in one draw is 3/4. Probability of not getting a club in n draws is (3/4)^n. We want 1 - (3/4)^n > 3/4, so (3/4)^n < 1/4. For n=4, (3/4)^4 = 81/256 = 0.316. For n=5, (3/4)^5 = 243/1024 = 0.237. Since 0.237 < 0.25, n=5 is the smallest integer.

AI explanation

The probability of not drawing a club in a single draw is 3/4. For n independent draws with replacement, the probability of getting no clubs is (3/4)^n. We need the probability of getting at least one club to be greater than 3/4, which means (3/4)^n must be less than 1/4. Testing n = 5, we get (3/4)^5 = 243/1024, which is less than 256/1024 (or 1/4), satisfying the inequality.

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