Question

A pair of dice is thrown $4$ times. If getting a double is considered a success, find the probability of two successes(approximately).

• A. $0.12$
• B. $0$
• C. $0.86$
• D. $1$

Answer 1

Answer: (A)

$0.12$

The repeated tosses of a pair of dice are Bernoulli trials.

Let $X$ denote the number of times of getting doublets in an experiment of throwing two dice simultaneously four times.
Probability of getting doublets in a single throw of the pair of dice is
$p=\frac{6}{36}=\frac{1}{6}$
$\therefore q=1-p=1-\frac{1}{6}=\frac{5}{6}$
Clearly, $X$ has the binomial distribution with $n=4,p=\frac{1}{6}$, and $q=\frac{5}{6}$
$\therefore P(X=x){=}^n{C}_x{q}^{n-x}{p}^x$, where $x=0,1,2,\mathrm{3...}n$
$={}^4{C}_x{\left(\frac{5}{6}\right)}^{4-x}\cdot {\left(\frac{1}{6}\right)}^x$
$={}^4{C}_x\cdot \frac{5^{4-x}}{6^4}$
$\therefore P\left(2\text{successes}\right)=P(X=2)$
$={}^4{C}_2\cdot \frac{5^{4-2}}{6^4}$
$=6\cdot \frac{25}{1296}$
$=\frac{25}{216}$

$=0.11576$

$\approx 0.12$