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Chapter 7: Problem 21

A card is drawn from a well-shuffled deck of 52 playing cards. Let \(E\) denote the event that the card drawn is black and let \(F\) denote the event that the card drawn is a spade. Determine whether \(E\) and \(F\) are independent events. Give an intuitive explanation for your answer.

Short Answer

Expert verified
Events \(E\) and \(F\) are independent because the probability of their intersection equals the product of their individual probabilities, that is, \(P(E \cap F) = P(E) \cdot P(F) = \frac{13}{52}\). Intuitively, the probability of drawing a spade is not affected by the fact that the card is black, as there's still a \(\frac{13}{52}\) chance of drawing a spade regardless of the card's color.

Step by step solution

01

Identify the probabilities of events E and F

There are 26 black cards in a standard deck of 52 playing cards (13 spades and 13 clubs), and there are 13 spades in total. To find the probability of each event, we'll divide the number of favorable outcomes by the total number of outcomes in the sample space. \(P(E) = \) Probability that the card drawn is black = \(\frac{26}{52}\) \(P(F) = \) Probability that the card drawn is a spade = \(\frac{13}{52}\)
02

Identify the probability of the intersection of events E and F

The intersection of events \(E\) and \(F\), denoted as \(E \cap F\), is the event that the card drawn is both black and a spade. Since all spades are black, this event is equivalent to the event that the card drawn is a spade. \(P(E \cap F) =\) Probability that the card drawn is black and a spade = \(\frac{13}{52}\)
03

Determine if E and F are independent

Events \(E\) and \(F\) are independent if the probability of their intersection equals the product of their individual probabilities: \(P(E) \cdot P(F) = \frac{26}{52} \cdot \frac{13}{52} = \frac{13}{52}\) Comparing, we have: \(P(E \cap F) = P(E) \cdot P(F)\)
04

Conclusion

Since the probabilities satisfy the condition of independence - the probability of the intersection of events E and F equals the product of their individual probabilities - events \(E\) and \(F\) are independent events. Intuitively, this makes sense because the probability of drawing a spade is not affected by the fact that the card is black; we would still have a 13 in 52 chance of drawing a spade regardless of the card's color.

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