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(i) Events are independent if the occurrence of one event does not affect the probability of the other event. In this case, the events of getting a card of hearts in the first draw and the second draw without replacement are dependent. The reason is that when a heart is drawn in the first draw, there is one less heart in the deck for the second draw, affecting the probability of getting a heart in the second draw.
(ii) To illustrate the probabilities of all possible outcomes of getting and not getting a card of heart in a tree diagram:
- Start with the first draw: Probability of getting a heart is 13/52 = 1/4, and the probability of not getting a heart is 39/52 = 3/4.
- For the second draw after not replacing the first card: If the first card was not a heart, the probability of getting a heart on the second draw is now 13/51, and the probability of not getting a heart on the second draw is 38/51. If the first card was a heart, the probabilities change accordingly.
(iii) To find the probability that both cards are hearts, you multiply the probabilities of drawing a heart in the first draw (1/4) and drawing a heart in the second draw given that the first card was a heart (12/51), as one heart is already drawn in the first pick. Multiply these probabilities: (1/4) * (12/51) = 3/51 = 1/17.
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