Question

A pair of dice is thrown 5 times. If the probability getting a doublet twice is P(A), then the value $3888P\left(A\right)=$ ___

Answer 1

Let us look at the experiment mentioned in the question. We are throwing a pair of dice and we are repeating that 5 times. The event we are interested in is getting a doublet two times.
When we throw a pair of dice, what is the probability of getting a doublet? Any pair of the form (x,x) is a doublet. There are 6 possible doublets and 36 possible outcomes when we throw a pair of dice. So the probability of getting a doublet would be $\frac{6}{36}=\frac{1}{6}$
We want to find the probability of getting doublet in 2 out of the 5 trials. This is same as finding probability of r successes in n trials, which would be equal to ${}^n{C}_r{p}^r{q}^{n-r}$, where p is the probability of success and q is the probability of failure. In our example, $p=\frac{1}{6}andq=\frac{5}{6}$
So the probability of 2 successes(two doublet) in 5 trials is given by ${}^5{C}_2{\left(\frac{1}{6}\right)}^2{\left(\frac{5}{6}\right)}^3$
$=\frac{625}{3888}$
$\Rightarrow$ 3888 P(A)= 625