Suppose during an international cycling competition 200 athletes are registered
ID: 3330657 • Letter: S
Question
Suppose during an international cycling competition 200 athletes are registered to
participate. One requirement is that all athletes should be cleared from taking any illegal aid
through medication for an unfair advantage. All the 200 athletes have obtained a certificate
declaring so but 12 of which have once or more than once tested positive in a doping test in
the past. To further ensure no wrongdoing is taking place a simple random sample of the
athletes undertake a fresh doping test after the opening ceremony. Let the random variable
X represents the number of athletes in the sample of size 12 that have tested positive to
doping.
(a) What is the probability that 3 of the athletes have blood samples that tested positive?
(b) What is the probability that at least one of the athletes has tested positive?
(c) Find the probability distribution of the athletes have tested positive in the past.
(d) Find mean, standard deviation, and variance.
(e) Also verify your results using formulae.
(f) Calculate the probability distribution using Matlab’s build in functions and also draw the
bar graph of the probability distribution using Matlab.
NOTE: PLEASE ATTEMPT ALL PARTS AND CLEARLY MENTION ALL STEPS. THANKYOU
Explanation / Answer
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
p = 12/200 = 0.06, N=12.
a)
P( X = 3 ) = ( 12 3 ) * ( 0.06^3) * ( 1 - 0.06 )^9
= 0.02723
b)
P( X < = 1) = P(X=1) + P(X=0) +
= ( 12 1 ) * 0.06^1 * ( 1- 0.06 ) ^11 + ( 12 0 ) * 0.06^0 * ( 1- 0.06 ) ^12 +
= 0.84046
P( X > 1) = 1 - P ( X <=1) = 1 -0.84046 = 0.15954
c) The probability distribution of the athletes have tested positive in the past is Binomial
d)
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 12 * 0.06
= 0.72
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 12 * 0.06 * 0.94
= 0.6768
III.standard deviation = sqrt( variance ) = sqrt(0.6768)
=0.82268