Respuesta :
ANSWER
P(X = x) = 0.401
EXPLANATION
We want to use Binomial Probability to find out the probability of obtaining exactly one 2 after rolling a fair die 6 times.
The Binomial Probaility method involves the use of the formula:
[tex]P(X=x)=^nC_x\cdot p^x\cdot q^{n\text{ - x}}[/tex]where n = number of trials in the experiment
x = number of successes in the experiment
p = probability of success in one trial
q = probability of failure in one trial
In this experiment, success would be getting a 2 in one roll of the die.
This means that, from the question:
n = 6
x = 1
p = 1/6 (probability of success in rolling a number from a die is always 1/6)
q = 5/6
Note: C means combination
So, we have that:
[tex]\begin{gathered} P(X=x)=^6C_1\cdot\text{ (}\frac{1}{6})^1\cdot\text{ (}\frac{5}{6})^{6\text{ - 1}} \\ P(X\text{ = x) = }\frac{6!}{(6\text{ - 1)!1!}}\cdot\text{ }\frac{1}{6}\cdot\text{ (}\frac{5}{6})^5 \\ P(X\text{ = x) = 6 }\cdot\text{ }\frac{1}{6}\cdot\text{ }\frac{3125}{7776} \\ P(X\text{ = x) = 0.40} \end{gathered}[/tex]That is the probability of rolling exactly one 2 after 6 times.
