Beck Evans
06/08/2023 · Middle School
Question 5 ( 30 marks) A card is drawn at random from a pack of 52 cards and is then replaced. Let \( A \) denote the event 'an ace is drawn,' and let \( R \) denote the event 'a red card is drawn.' Calculate the values of \( P(A), P(R), P(A \) \( n \mathrm{R}) \) and \( P(A \) u R). If this experiment is performed 4 times, find the probability i. That an ace will be drawn for the first time at the third attempt. ii. That a red card will be drawn exactly twice. iii. That a red card will be drawn at least once. Find also how many times the experiment must be performed to ensure that the probability of a red card being drawn at least once exceeds 0.99 .
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1. \( P(A) = \frac{1}{13} \), \( P(R) = \frac{1}{2} \), \( P(A \cap R) = \frac{1}{26} \), \( P(A \cup R) = \frac{25}{26} \)
2. \( P(\text{Ace at 3rd attempt}) = \frac{144}{2197} \)
3. \( P(\text{Red card twice}) = \frac{3}{8} \)
4. \( P(\text{Red card at least once}) = \frac{15}{16} \)
5. At least 7 experiments to exceed 0.99 probability of drawing a red card.
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