There are 2 6-sided dice. The first die is a normal fair die where each face has

Question

There are 2 6-sided dice. The first die is a normal fair die where each face has a probability of showing of 1/6. The second die is biased so that the probability of showing the largest face 6 is twice as high as of the other faces, and all of the other faces have equal probabilities.

1) What is the expected value of the outcome if a fair 6-sided die is thrown?

2) What is the expected value of the outcome from tossing the biased die?

If you throw the fair die 4 times, then the biased die 2 times consecutively and sum up all the outcomes:

3) What is the expected value of the sum?

Let Y denote the sum from the previous part. If we know that

P(Y > k) 0.4

4) According to Markov's inequality, what is k?

Explanation / Answer

(1) expected value=sum(x*p(x))=3.5

(2) here p(6)=2a and P(1)=p(2)=p(3)=p(4)=p(5)=a

so, 7a=1 and a=1/7

Expected value=3.86

(3) expected sum=4*3.5+2*3.86= 21.72

(4) markov inequality state that P[Yk]E[Y]/k

E(Y)=21.72 and P(Y>k)=0.4

so k=0.4*21.72=8.688

x 1 2 3 4 5 6 sum p(x) 0.17 0.17 0.17 0.17 0.17 0.17 1.00 x*p(x) 0.166667 0.333333 0.5 0.666667 0.833333 1 3.50

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