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Chapter 7: Problem 40

Two cards are drawn without replacement from a wellshuffled deck of 52 cards. Let \(A\) be the event that the first card drawn is a heart, and let \(B\) be the event that the second card drawn is a red card. Show that the events \(A\) and \(B\) are dependent events.

Short Answer

Expert verified
Events \(A\) and \(B\) are dependent events because the probability of both events happening together, \(P(A \cap B) = \frac{1}{8}\), is not equal to the product of the individual probabilities, \(P(A) \cdot P(B) = \frac{13}{102}\).

Step by step solution

01

Calculate probabilities of individual events

First, let's find the probability of each individual event. For event \(A\), the probability of drawing a heart as the first card is equal to the number of hearts divided by the total number of cards in the deck. There are 13 hearts in a deck of 52 cards, so: \(P(A) = \frac{13}{52} = \frac{1}{4}\) Next, let's find the probability of Event \(B\). A red card can either be a heart or a diamond, so there are 26 red cards in total. Since we have already drawn one card (referenced in Event A), there are now 51 cards left. \(P(B) = \frac{26}{51}\)
02

Calculate the probability of both events happening together

Now, we need to find the probability of both events (\(A\) and \(B\)) happening together. If the first card drawn is a heart, there are 12 hearts left in the deck with 51 cards remaining. For the second card to be a red card (which includes hearts and diamonds), there are 25 red cards left. So, the probability of both events happening together is: \(P(A \cap B) = \frac{12}{51} \cdot \frac{25}{50}\) \(P(A \cap B) = \frac{1}{4} \cdot \frac{1}{2}\) \(P(A \cap B) = \frac{1}{8}\)
03

Check if events \(A\) and \(B\) are dependent

Finally, let's check if events \(A\) and \(B\) are dependent by comparing the product of the individual probabilities with the probability of both events happening together. If these two values are not equal, then the events are dependent. \(P(A) \cdot P(B) = \frac{1}{4} \cdot \frac{26}{51} = \frac{13}{102}\) Since \(P(A \cap B) = \frac{1}{8}\) and \(P(A) \cdot P(B) = \frac{13}{102}\), we can conclude that: \(P(A \cap B) \neq P(A) \cdot P(B)\) This tells us that events \(A\) and \(B\) are dependent events.

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