One card is drawn from a deck. Are the events "drawing a face card" and "drawing a black card" independent or dependent?
Are there black face cards?
Yes, but of the faces, 1/2 are black. So 1/2 means independent.
To determine whether the events "drawing a face card" and "drawing a black card" are independent or dependent, we need to understand the definitions of these terms.
- A face card refers to any card in a deck that has a face on it, namely, the Jack (J), Queen (Q), and King (K) cards in each suit. There are a total of 12 face cards in a deck (3 face cards per suit).
- A black card refers to any card in a deck that has a black suit, namely, the Spades (♠) and Clubs (♣) suits. There are a total of 26 black cards in a deck (13 black cards per suit).
Now, let's analyze the dependency of the events:
If these events are independent, it means that the outcome of one event does not affect the likelihood of the other event occurring. In other words, drawing a face card does not affect the probability of drawing a black card.
To determine the probability of drawing a face card, we divide the number of favorable outcomes (12 face cards) by the total number of possible outcomes (52 cards in a deck). Therefore, the probability of drawing a face card is 12/52, which simplifies to 3/13.
To determine the probability of drawing a black card, we divide the number of favorable outcomes (26 black cards) by the total number of possible outcomes (52 cards in a deck). Therefore, the probability of drawing a black card is 26/52, which simplifies to 1/2.
Since the probability of drawing a face card (3/13) is different from the probability of drawing a black card (1/2), we can conclude that the events "drawing a face card" and "drawing a black card" are dependent.
In summary, drawing a face card and drawing a black card are dependent events because the outcome of one event (drawing a face card) affects the likelihood of the other event (drawing a black card) occurring.