Questions: A card is drawn from a well-shuffled deck of 52 cards. Find the probability that the card is a black card or a 3.

Solution Steps

To find the probability of drawing a black card or a 3 from a deck of 52 cards, we can use the principle of inclusion-exclusion. First, calculate the probability of drawing a black card, then the probability of drawing a 3, and finally subtract the probability of drawing a card that is both black and a 3, as these have been counted twice.

A standard deck of 52 cards contains 26 black cards (13 spades and 13 clubs). The probability of drawing a black card is given by: \[ P(\text{Black}) = \frac{26}{52} = 0.5 \]

There are 4 cards with the rank of 3 in a deck (one for each suit). The probability of drawing a 3 is: \[ P(3) = \frac{4}{52} \approx 0.07692 \]

Among the 4 threes, 2 are black (3 of spades and 3 of clubs). The probability of drawing a black 3 is: \[ P(\text{Black 3}) = \frac{2}{52} \approx 0.03846 \]

To find the probability of drawing a black card or a 3, we use the inclusion-exclusion principle: \[ P(\text{Black or 3}) = P(\text{Black}) + P(3) - P(\text{Black 3}) \] Substituting the values, we get: \[ P(\text{Black or 3}) = 0.5 + 0.07692 - 0.03846 \approx 0.5385 \]

Final Answer