State whether the events described are independent or dependent and determine the probability:
A card is pulled from a well shuffled deck of standard playing cards and is not put back in the deck. A second card is drawn from the deck. What is the probability that a spade was drawn both times?
To determine the probability, we need to identify whether the events described are independent or dependent.
In this case, the events are dependent because the first card drawn is not put back into the deck before the second card is drawn. The probability of drawing a spade on the second draw is influenced by the outcome of the first draw.
To calculate the probability, we need to consider that the deck initially contains 52 cards, and 13 of them are spades.
The probability of drawing a spade on the first draw is P(S1) = 13/52 = 1/4 (since there are 13 spades out of 52 cards).
After drawing a spade on the first draw, there are 51 cards left in the deck, and 12 of them are spades. So, the probability of drawing a spade on the second draw (given that a spade was drawn on the first draw) is P(S2|S1) = 12/51.
To calculate the overall probability of drawing a spade both times, we multiply the probabilities of the individual events:
P(S1 and S2) = P(S1) × P(S2|S1) = (1/4) × (12/51) = 3/51 = 1/17.
Therefore, the probability of drawing a spade on both draws is 1/17.