Question

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that:
(i) all the five cards are spades.
(ii) none is spade.

Hint:

For drawing cards with replacement, use the multiplication rule of probability, raising the individual probability to the power of the number of trials.

Answer 1

Step 1: Find the probability of drawing a spade.
A standard deck of 52 cards has 13 spades. So, the probability of drawing a spade in one draw is: \[ P(\text{spade}) = \frac{13}{52} = \frac{1}{4} \] The probability of not drawing a spade is: \[ P(\text{not spade}) = 1 - \frac{1}{4} = \frac{3}{4} \]

Step 2: Probability that all five cards are spades.
Since the cards are drawn with replacement, the probability that all five cards drawn are spades is: \[ P(\text{all five spades}) = \left( \frac{1}{4} \right)^5 = \frac{1}{1024} \]

Step 3: Probability that none of the five cards is a spade.
The probability that none of the five cards drawn is a spade is: \[ P(\text{none is spade}) = \left( \frac{3}{4} \right)^5 = \frac{243}{1024} \]

Final Answer: (i) Probability that all five cards are spades: \[ \boxed{\frac{1}{1024}} \] (ii) Probability that none is a spade: \[ \boxed{\frac{243}{1024}} \]