Question

A fair six-sided dice (with sides 1, 2, ...,6) is rolled 4 times. Find the probability that at least one 2 is rolled. Give your answer as an exact fraction.

Answer 1

We can model this with a binomial random variable, with sample size n=4 and probability of success p=1/6.

As we have to calculate the probability of getting at least one 2, it is easy to substract from a probability equal to 1 the probability of getting no 2. This can be written as:

`\begin{gathered} P(k\ge1)=1-P(k=0)=1-(1-p)^n \\ P(k\ge1)=1-(1-(1)/(6))^4=1-((5)/(6))^4=1-(625)/(1296)=(1296-625)/(1296)=(671)/(1296) \end{gathered}`

The probability of getting at least one 2 in dice rolled 4 times is P=671/1296.