Question

Cards are drawn at random and with replacement from an ordinary deck of 52 cards until a spade appears. (a) What is the probability that at least four draws are necessary? (b) Same as part (a), except the cards are drawn without replacement.

Short answer

The probability that at least four draws are necessary with replacement is \((\frac{3}{4})^3\) and without replacement is \(\frac{39}{52} \times \frac{38}{51} \times \frac{37}{50}\).

Step-by-step solution

Solving question (a) - with replacement

The total number of cards is 52, with 13 of those being spades. Thus, probability of drawing a spade is \(\frac{13}{52}\) or \(\frac{1}{4}\) and the probability of not drawing a spade is \(\frac{39}{52}\) or \(\frac{3}{4}\). We are asked to find probability that at least four draws are necessary to get a spade. This is equivalent to the first three draws not being a spade. Since the draws are with replacement, each draw is independent and therefore the probabilities multiply. Probability of at least four draws needed = \( (\frac{3}{4})^3 \).

Solving question (b) - without replacement

In the scenario without replacement, the total number of cards decreases every draw, as does the number of non-spades. So, the probability of the first card not being a spade is \(\frac{39}{52}\), of the second card not being a spade is \(\frac{38}{51}\), and of the third card not being a spade is \(\frac{37}{50}\). So, the probability of needing at least four draws to get a spade is \(\frac{39}{52} \times \frac{38}{51} \times \frac{37}{50}\).