Question

a deck of 40 cards contains 20 blue and 20 red cards. each color has cards numbered 1 to 20. which events are independent? choose all that apply. a. a 2 is chosen from the deck. the card is put back into the deck and then a 2 is chosen. b. a 2 is chosen from the deck. the card is not put back into the deck and then a 2 is chosen. c. a red 5 is chosen from the deck. the card is put back into the deck and then a red card is chosen. d. a red 5 is chosen from the deck. the card is not put back into the deck and then a red card is chosen. e. a red card is chosen from the deck. the card is put back into the deck and then a blue card is chosen. f. a red card is chosen from the deck. the card is not put back into the deck and then a blue card is chosen.

Explanation:

Step1: Recall definition of independent events

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\), which means the occurrence of one event does not affect the probability of the other event. This is true when sampling is done with - replacement.

Step2: Analyze option A

The first - draw is a 2, and since the card is put back, the probability of drawing a 2 on the second draw is not affected by the first draw. Let \(A\) be the event of drawing a 2 on the first draw and \(B\) be the event of drawing a 2 on the second draw. \(P(A)=\frac{2}{40}\) (2 cards numbered 2 in 40 cards), and \(P(B)=\frac{2}{40}\) (because the card is replaced), and \(P(A\cap B)=P(A)\times P(B)\), so this is an independent event.

Step3: Analyze option B

The first - draw is a 2, and since the card is not put back, the number of cards and the number of 2 - cards change for the second draw. So the probability of drawing a 2 on the second draw is affected by the first draw, and these are dependent events.

Step4: Analyze option C

The first - draw is a red 5. Since the card is put back, the probability of drawing a red card on the second draw is not affected by the first draw. Let \(A\) be the event of drawing a red 5 on the first draw (\(P(A)=\frac{1}{40}\)) and \(B\) be the event of drawing a red card on the second draw (\(P(B)=\frac{20}{40}\)). Since the card is replaced, \(P(A\cap B)=P(A)\times P(B)\), so this is an independent event.

Step5: Analyze option D

The first - draw is a red 5. Since the card is not put back, the number of red cards and the total number of cards change for the second draw. So the probability of drawing a red card on the second draw is affected by the first draw, and these are dependent events.

Step6: Analyze option E

The first - draw is a red card. Since the card is put back, the probability of drawing a blue card on the second draw is not affected by the first draw. Let \(A\) be the event of drawing a red card on the first draw (\(P(A)=\frac{20}{40}\)) and \(B\) be the event of drawing a blue card on the second draw (\(P(B)=\frac{20}{40}\)). Since the card is replaced, \(P(A\cap B)=P(A)\times P(B)\), so this is an independent event.

Step7: Analyze option F

The first - draw is a red card. Since the card is not put back, the number of red and blue cards and the total number of cards change for the second draw. So the probability of drawing a blue card on the second draw is affected by the first draw, and these are dependent events.

Answer:

A. A 2 is chosen from the deck. The card is put back into the deck and then a 2 is chosen.
C. A red 5 is chosen from the deck. The card is put back into the deck and then a red card is chosen.
E. A red card is chosen from the deck. The card is put back into the deck and then a blue card is chosen.