Question

In throwing a fair die, following are the probabilities of getting each face.
1 - k
2 - 2k
3 - 2k
4 - 3k
5 - $3k^2$
6 - $7k^2+k$
Expected value of the outcome = ___

Answer 1

This is a continuation of the previous problem we solved. We want to find the expected value of outcome. It would be given by 1.p(1) + 2.p(2) + 3.p(3) + 4. p(4) + 5. p(5) + 6. p(6)
where p(i) is the probability of getting face ‘i’
$\Rightarrow expectedvalue=(1\times k)+(2\times 2k)+(3\times 2k)+(4\times 3k)+(5\times 3k^2)+\left(6\right(7k^2+k\left)\right)=k+4k+6k+12k+15k^2+42k^2+6k=29k+57k^2$
If we add all the probabilities we should get 1, because the probability of sample space is 1.
$\Rightarrow$ k+2k+2k+ 3k+$3k^2$+(7k^2 + k) = 1
$\Rightarrow$ $10k^2$ + 9k - 1 = 0
$\Rightarrow$ k = -2 or $k=\frac{1}{10}$
Since probability can’t be negative $k=\frac{1}{10}$
$\Rightarrow$ Expectation or mean value $=29\times \frac{1}{10}+57\times {\left(\frac{1}{10}\right)}^2$
$=\frac{347}{100}$
= 3.47