Question

A fair die is rolled 5 times. What is the probability that a 2 is obtained on at least one of the rolls? Round your answer to three decimal places. (If necessary, consult a list of formulas.) 

Answer 1

Solution:-

Suppose a fair die is rolled n=5 times.

The possible outcomes of the die are

S={1,2,3,4,5,6}

Since, the die is fair, thus the probability that any of number from set S is 1/6

Let

X: The number of times "2" is occured on face of die

p: probability of success corresponding to X

p=1/6= probability of getting 2 on die at any rolls

Therefore,

From above information,

X~ Binomial (n=5,p=1/6) distribution

The pmf of X is given by

$P(X=x) =\left\{\begin{matrix}\binom{n}{x}p^x(1-p)^{n-x} & ;x=0,1,2,....n\\ &0<p<1,n>0\\0 & ; Otherwise \end{matrix}\right.$

Now, the question is,

What is probability that 2 is obtained on atleast one of the rolls?

----> that is we have to find,

P(X>=1) =?

P(X>=1 ) = 1 - P(X<1)

P(X>=1) = 1 - P(X=0)

From pmf of X,

$P(X\geq 1)=1- \binom{5}{0}(\frac{1}{6})^0(1-\frac{1}{6})^{5-0}$

$P(X\geq 1)=1- 1*1*(\frac{5}{6})^{5}$

$P(X\geq 1)=1- 0.401877572$

PX > 1) = 0.598122428

$P(X\geq 1)=0.5981---(rounded\_ to\_ 4\_ decimal)$

Result:-

The probability that a 2 is obtained on atleast one of the rolls is 0.5981