Final answer:
Calculating the probability of specific card combinations requires understanding the rules of combinatorics. For each scenario, one must consider the sequence of events and apply the principles of probability without replacement. The result offers the likelihood of the given set of outcomes in a deck of cards.
Step-by-step explanation:
When dealing with probabilities in a deck of cards, you must carefully consider each scenario and calculate the likelihood using combinatorial principles.
(a) First club is the third card dealt:
To find the probability that the first club you get is the third card dealt, consider the sequence of events. The first two cards must not be clubs. There are 39 non-club cards, so the probability of drawing a non-club first is 39/52. Without replacement, the probability of drawing a second non-club is 38/51. Finally, with only clubs remaining to be our first club, the probability for the third card being a club is 13/50. Multiply these probabilities to find the total probability: (39/52) * (38/51) * (13/50) = 0.0551, or 5.51%.
(b) All cards are diamonds:
With 13 diamonds in the deck, the probability of drawing a diamond first is 13/52. Then, without replacement, the probability of drawing another diamond is 12/51, and for a third is 11/50. Multiply these together: (13/52) * (12/51) * (11/50) = 0.0129, or 1.29%.
(c) No aces are drawn:
The deck has four aces. To draw no aces, we must draw from the 48 other cards each time. The probability is (48/52) * (47/51) * (46/50) = 0.7820, or 78.20%.
(d) At least one spade in the hand:
The probability of at least one spade can be found by subtracting the probability of drawing no spades from 1. The probability of drawing no spades (there are 39 non-spades) three times in a row is (39/52) * (38/51) * (37/50). Therefore, the probability of at least one spade is 1 - (39/52) * (38/51) * (37/50) = 0.6618 or 66.18%.