Tennis Set Tiebreaker
Three Statistics questions?
1. According to police sources a car with a certain protection system will be recovered 86% of the time. Find the probability that 4 of 9 stolen cars will be recovered.
2. Sixty-five percent of men consider themselves knowledgeable football fans. If 11 men are randomly selected, find the probability that exactly four of them will consider themselves knowledgeable fans.
3. The probability that a tennis set will go to a tiebreaker is 11%. In 240 randomly selected tennis sets, what is the mean and the standard deviation of the number of tiebreakers?
Let X be the number of cars recovered. X has the binomial distribution with n = 9 trials and success probability p = 0.86.
In general, if X has the binomial distribution with n trials and a success probability of p then
P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)
for values of x = 0, 1, 2, …, n
P[X = x] = 0 for any other value of x.
this is found by looking at the number of combination of x objects chosen from n objects and then a total of x success and n – x failures.
Or, to be more accurate, the binomial is the sum of n independent and identically distributed Bernoulli trials.
P(X = 0 ) = 2.066105e-08
P(X = 1 ) = 1.142261e-06
P(X = 2 ) = 2.806698e-05
P(X = 3 ) = 0.0004022934
P(X = 4 ) = 0.003706846 < === ANSWER
P(X = 5 ) = 0.02277062
P(X = 6 ) = 0.09325113
P(X = 7 ) = 0.2454979
P(X = 8 ) = 0.3770146
P(X = 9 ) = 0.2573274
//// ==== Question 2 ==== \\
Let M be the number of men considering themselves knowledgeable football fans. M as the binomial distribution with n = 11 trials and success probability p = 0.65.
P(M = 0 ) = 9.654916e-06
P(M = 1 ) = 0.0001972361
P(M = 2 ) = 0.001831478
P(M = 3 ) = 0.01020395
P(M = 4 ) = 0.03790039 < === ANSWER
P(M = 5 ) = 0.09854101
P(M = 6 ) = 0.1830047
P(M = 7 ) = 0.2427614
P(M = 8 ) = 0.2254213
P(M = 9 ) = 0.1395465
P(M = 10 ) = 0.05183156
P(M = 11 ) = 0.008750783
//// ==== Question 3 ==== \\
the mean of the binomial distribution is n * p
the variance of the binomial distribution is n * p * (1 - p)
the standard deviation is the square root of the variance.
Let T be the number of tennis matches that go to a tiebreaker.
T has the binomial distribution with n = 240 trials and success probability p = 0.11
Mean of T is 240 * 0.11 = 26.4
Variance of T is 240 * 0.11 * (1 - 0.11) = 23.496
Standard Deviation = sqrt(23.496) = 4.847267
