This weekend, Premier League officials will be conducting urine tests for all players who are participating in Premier League matches. The tests are being carried out as part of the league's ongoing efforts to combat the use of performance-enhancing drugs and other banned substances. The urine samples will be collected before and after the matches, and will be analysed by a World Anti-Doping Agency (WADA)-accredited laboratory to detect any traces of banned substances in the players' systems. Suppose 3% of the players will use banned substances. Let X be the number of players that officials need to test to find the first case of using banned substances. Let Y be the number of players tested to find 3 such cases. (a) Name the probability distribution and specify the value of any parameter(s) for each of the two random variables X and Y. (b) What is the probability that at least 5 players are to be tested to find the first using-banned-substances case? (c) What is the probability that exactly 30 players are to be tested to find 3 using- banned-substances cases? (d) On average, how many players do officials need to test to find 3 cases of using banned substances? (e) Find P(Y > 50). (f) Suppose the officials have not found any case of using banned substances in their first 5 tests. What is the conditional probability that the official still cannot find any case in their next 5 tests? (g) What is the probability that the officials will find exactly 4 cases of using banned substances in testing 100 players? At least two different probability distributions can be used to calculate/approximate 100 1 111
This weekend, Premier League officials will be conducting urine tests for all players who are participating in Premier League matches. The tests are being carried out as part of the league's ongoing efforts to combat the use of performance-enhancing drugs and other banned substances. The urine samples will be collected before and after the matches, and will be analysed by a World Anti-Doping Agency (WADA)-accredited laboratory to detect any traces of banned substances in the players' systems. Suppose 3% of the players will use banned substances. Let X be the number of players that officials need to test to find the first case of using banned substances. Let Y be the number of players tested to find 3 such cases. (a) Name the probability distribution and specify the value of any parameter(s) for each of the two random variables X and Y. (b) What is the probability that at least 5 players are to be tested to find the first using-banned-substances case? (c) What is the probability that exactly 30 players are to be tested to find 3 using- banned-substances cases? (d) On average, how many players do officials need to test to find 3 cases of using banned substances? (e) Find P(Y > 50). (f) Suppose the officials have not found any case of using banned substances in their first 5 tests. What is the conditional probability that the official still cannot find any case in their next 5 tests? (g) What is the probability that the officials will find exactly 4 cases of using banned substances in testing 100 players? At least two different probability distributions can be used to calculate/approximate 100 1 111
Chapter9: Sequences, Probability And Counting Theory
Section9.7: Probability
Problem 3SE: What is an experiment?
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Transcribed Image Text:This weekend, Premier League officials will be conducting urine tests for all players
who are participating in Premier League matches. The tests are being carried out as
part of the league's ongoing efforts to combat the use of performance-enhancing drugs
and other banned substances. The urine samples will be collected before and after the
matches, and will be analysed by a World Anti-Doping Agency (WADA)-accredited
laboratory to detect any traces of banned substances in the players' systems.
Suppose 3% of the players will use banned substances. Let X be the number of players
that officials need to test to find the first case of using banned substances. Let Y be
the number of players tested to find 3 such cases.
(a) Name the probability distribution and specify the value of any parameter(s) for
each of the two random variables X and Y.
(b) What is the probability that at least 5 players are to be tested to find the first
using-banned-substances case?
(c) What is the probability that exactly 30 players are to be tested to find 3 using-
banned-substances cases?
(d) On average, how many players do officials need to test to find 3 cases of using
banned substances?
(e) Find P(Y > 50).
(f) Suppose the officials have not found any case of using banned substances in their
first 5 tests. What is the conditional probability that the official still cannot find
any case in their next 5 tests?
(g) What is the probability that the officials will find exactly 4 cases of using banned
substances in testing 100 players?
At least two different probability distributions can be used to calculate/approximate
this probability. Which two, and how different are the results (in terms of absolute
difference)?
What if we change 100 players mentioned above to 400 players?
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