31. Sephen is practising basketball penalty shots. The probability that he scores in the firet attempt is 0.75 and each subsequent attempt that he makes is affected by the previous one. The probability that he scores in thenext penalty shot is 0.8 if he scores in the previous one. The probabiliny that he scores in the next penalty shotis 0.45 if he does not score in the previous one. Find the probability that (a) he scores in the first two shots. B.he scores at least once in the first two shots, C.he does not score until his fourth shot. 34. Terry puts three different letters randomly in three envelopes with different addresses, Find the probability thatat least one letter is put in the envelope with correct address.
Question
- Sephen is practising basketball penalty shots. The probability that he scores in the firet attempt is 0.75 and each subsequent attempt that he makes is affected by the previous one. The probability that he scores in thenext penalty shot is 0.8 if he scores in the previous one. The probabiliny that he scores in the next penalty shotis 0.45 if he does not score in the previous one. Find the probability that (a) he scores in the first two shots. B.he scores at least once in the first two shots, C.he does not score until his fourth shot.
- Terry puts three different letters randomly in three envelopes with different addresses, Find the probability thatat least one letter is put in the envelope with correct address.
Solution
Sure, let's solve these problems one by one.
(a) The probability that he scores in the first two shots: The probability that he scores in the first shot is 0.75. Given that he scores in the first shot, the probability that he scores in the second shot is 0.8. Therefore, the probability that he scores in the first two shots is 0.75 * 0.8 = 0.6.
(b) The probability that he scores at least once in the first two shots: This is the complement of the event that he does not score in both shots. The probability that he does not score in the first shot is 1 - 0.75 = 0.25. Given that he does not score in the first shot, the probability that he does not score in the second shot is 1 - 0.45 = 0.55. Therefore, the probability that he does not score in both shots is 0.25 * 0.55 = 0.1375. Hence, the probability that he scores at least once in the first two shots is 1 - 0.1375 = 0.8625.
(c) The probability that he does not score until his fourth shot: This is the probability that he does not score in the first three shots and then scores in the fourth shot. The probability that he does not score in the first shot is 0.25. Given that he does not score in the first shot, the probability that he does not score in the second shot is 0.55. Given that he does not score in the second shot, the probability that he does not score in the third shot is again 0.55. The probability that he scores in the fourth shot given that he did not score in the third shot is 0.45. Therefore, the probability that he does not score until his fourth shot is 0.25 * 0.55 * 0.55 * 0.45 = 0.03421875.
For the last question, Terry puts three different letters randomly in three envelopes with different addresses. The total number of ways to put the letters in the envelopes is 3!, which is 6. The event that at least one letter is put in the envelope with the correct address is the complement of the event that no letter is put in the correct envelope. There are only two ways to put the letters in the envelopes such that no letter is in the correct envelope, which are the derangements of three elements. Therefore, the probability that at least one letter is put in the envelope with the correct address is 1 - 2/6 = 2/3.
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