2. (a) There was an exam where 10000 students took, and the average was 40 with

Question

2.

(a) There was an exam where 10000 students took, and the average was 40 with variance being 10. What can you say about the number of students that scored above 80? (Use both Markov and Chebyshev and compare the results)

(b) We are going to flip a fair coin 100 times. What can you say about the probability that we will get at least 70 heads?

(c) There is a basketball player that has scored 1200 points over the course of 60 games, with the variance being 5. What can you say about the number of games where he scored at least 30 points? (Use both Markov and Chebyshev, and compare the results)

Explanation / Answer

Markov's inequality states P( X >=r) <= E (X)/ r

Chebyshev's inequality states P ( |X- E(X)| >= r) <= V(X)/ r sqr

a) E(X) = mean = 40, V(X) = 10

Markov P (X>= 80) <= 40/80 i.e. 1/2

Chebychev P( |X-40|>=r) <= 10/ r^2

We know x - 40 is 80, hence r is 120

P ( |X-40| >=120) <= 10/120^2 = 1/240

i.e. 0.000694

b) Markov

P(X >=70) <= (1/2*100)/70 = 50/70 = 0.7143

Chebychev

P (|X - 50| >= r) <= V(X)/r sq

we know x -50 is 70 hence r is 120

P ( |x-50| >=120) <= (1/2*1/2*100)/120^2

i.e. 0.00173

c) Mean = E(X) = 1200/60 = 20, V(X) = 5

markov - P (X>=30) <= 20/30

i.e. probability of scoring over 30 pts every game is atmost 2/3

If he has played 60 games, the number of games he has scored more than 30 would be a maximum of 40 (2/3*60)

Chebychev - P( |X-20| > r) <= 5/ r sq

hence r is 50

P ( |X-20| >= 30) <= 5/30^2 = 0.0056

i.e. If he has played 60 there might be a maximum of 0.33 games (i.e. 1 game it at all) where his score would have been over 30 points

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