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B rolls a die and scores the number obtained on the roll. A rolls two dice and scores the larger number of the two dice. whoever gets lager number win $$X=\begin{cases} 100,000&\text{A wins}\\ -x&\text{A loses}\\ 0&\text{draw}\end{cases}$$

  1. Find the probability A wins with $6$
  2. Find the probability A wins
  3. Find the probability B win
  4. Find the probability of a draw
  5. Find $x$ for which $E[X]=0$

for A and B
A B
1 1/36 1/6
2 3/36 1/6
3 5/36 1/6
4 7/36 1/6
5 1/4 1/6
6 11/36 1/6
the idea here for example biggest dice number that A roll is 2 so (1,2),(2,1),(2,2) so 3/36 and the rest same.

1 probability A wins at 6 : $11/36∗5/6=55/216$
the idea is to multiply probability A win at k and and probability B have dice number smaller than k

2 probability A win : A win at 2+A win at 3+A win at 4+ A win at 5+A win at 6 $=1/72+5/108+7/72+1/6+55/216$

probability of B win
B win at 2+B win at 3+B win at 4+B win at 5+B win at 6
1/216+1/54+1/24+16/216+25/216

Probability draw
draw at 1+ draw at 2 + draw at 3+ draw at 4+ draw at 5+ draw at 6 br> = $1/216+1/72+5/215+7/216+1/24+11/216$

5.how to find x so $E[X]=0$

I'm not quite sure for this one, but what is the relation of this question with the random variable X? am i right?

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  • $\begingroup$ What do you mean by the probability that $E[X]=0$? That's not a question of probability. It's $0$ or it isn't. $\endgroup$ – saulspatz May 28 at 15:15
  • $\begingroup$ @saulspatz im sorry i need to find x in order E[X] = 0 $\endgroup$ – devss May 28 at 15:26
  • $\begingroup$ Please edit your question to correct it. Also, do you understand how to compute expectation? I'm not sure what your problem is. $\endgroup$ – saulspatz May 28 at 15:36
  • $\begingroup$ @saulspatz yes i already corrected it, but here i dont understand the relation of random variable $X$ to this question, is it to compute expectation? $\endgroup$ – devss May 28 at 15:38
  • $\begingroup$ @DavidK i wrote it when i first asked this question ,whoever gets larger number win ,but someone edited it and somewhat all important things is deleted :/ $\endgroup$ – devss May 29 at 3:23
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$$E(X)= 100000P(A)-xP(B)$$ by definition of expectation, where $P(A)$ is the probability that $A$ wins, and $P(B)$ is the probability that $B$ wins. You have only to substitute the values that you computed in steps 1 and 2 and solve for $x$.

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