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Q.
A pair of dice is thrown $200$ times. If getting a sum of $9$ is considered a success, then find the mean and the variance respectively of the number of successes.
Probability - Part 2
Solution:
When a pair of dice is thrown, the sample space has $36$ equally likely outcomes.
Outcomes favourable to sum of $9$ are $(5,\, 4)$, $(4,\, 5)$,
$(6,\,3)$, $(3,\,6)$ only.
So $p =$ probability (sum is $9$) $= \frac{4}{36} = \frac{1}{9}$,
$\therefore q = 1-\frac{1}{9}= \frac{8}{9}$, $n = 200$
Hence, mean $= np= 200\left(\frac{1}{9}\right)= \frac{200}{9}$ and
variance $= npq = 200\left(\frac{1}{9}\right)\left(\frac{8}{9}\right) = \frac{1600}{81}$