# A fair die is rolled 12 times. What is the

A fair die is rolled 12 times. What is the probability of rolling a 4 on the die 4 times?
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Step 1
A binomial probability distribution has following properties.
1. There are only 2 possible outcomes of an experiment.
2. The total number of trials are fixed.
3. The probability of success in each trial is equal.
4. Each trial is independent to other.
The formula to find the probability of binomial random variable X is
$P\left(X\right)=\frac{n!}{x!\left(n-x\right)!}{p}^{x}{\left(1-p\right)}^{n-x}$, where x is the number of successful trials, n is the total number of trials and p is the probability of success in each trial.
Step 2
If a die is rolled then there are total 6 possible outcomes {1, 2, 3, 4, 5, 6} and only 1 outcome is favourable for getting a 4 on die. So the probability that a 4 comes when die is rolled will be ratio of 1 favourable outcomes to total 6 possible outcomes.
$p=\frac{1}{6}$
For the given experiment, there will be successful trial if die gives a 4 otherwise unsuccessful. So two outcomes will be considered either a 4 comes or not. The die is rolled total 12 times so the total number of trials are fixed and among those 12 trials 4 should be successful. Each time the die is rolled is independent to previous outcomes, so each trial is independent and probability of getting a 4 will be equal in each trial.
Step 3
As it satisfies all conditions of binomial distribution, the probability that $x=4$ trials give a 4 out of $n=12$ trials can be found using the formula $P\left(X\right)=\frac{n!}{x!\left(n-x\right)!}{p}^{x}{\left(1-p\right)}^{n-x}$
Substitute in the above mentioned formula.
$P\left(4\right)=\frac{12!}{4!\left(12-4\right)!}{\left(\frac{1}{6}\right)}^{4}{\left(1-\frac{1}{6}\right)}^{12-4}$
$=\frac{12\cdot 11\cdot 10\cdot 9\cdot 8!}{24\cdot 8!}\left(\frac{1}{1296}\right){\left(\frac{5}{6}\right)}^{8}$
$=495\cdot \left(\frac{1}{1296}\right)\cdot \left(\frac{390625}{1679616}\right)$
$\approx 0.0888$
So the probability of rolling a 4, four times out of 12 trials will be 0.0888.