Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. (i) What is the probability that the card is the queen? (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?
Step-by-Step Solution
Key Concept: Use the definition of probability: \(P(E)=\dfrac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}\). After drawing the queen and setting it aside, the sample space reduces, and the new probabilities are computed with the reduced total.
1. Total number of cards: There are 5 cards (10, J, Q, K, A). All are equally likely to be drawn.\
2. Part (i): \(\text{Favourable outcomes}=1\) (only the queen). Hence\
$$P(\text{queen}) = \frac{1}{5}.$$\
3. Part (ii) – The queen has been drawn and removed. The remaining cards are 10, J, K, A (4 cards).\
- (a) Probability of drawing an ace: Only one ace remains among the 4 cards.\
$$P(\text{ace}\mid\text{queen removed}) = \frac{1}{4}.$$\
- (b) Probability of drawing a queen: The queen is no longer in the deck, so there are 0 favourable outcomes.\
$$P(\text{queen}\mid\text{queen removed}) = \frac{0}{4}=0.$$
Correct Answer: (i) \(\dfrac{1}{5}\)\; (ii)(a) \(\dfrac{1}{4}\)\; (ii)(b) \(0\)