Questions: You roll a fair die 18 times; the rolls are independent of each other. What is the probability that you roll a 5 exactly three times?
Answer: 0.245198448
Attempt: From the basics, I have learned so far, I need to determine the sample space and the event for this case and then do event/sample space
Sample Space: {1,2,3,4,5,6}*18 = 1/6*18=3
Event: Roll a 5 exactly 3 times, so 1/6*3=1/2
Pr(A) = 1/2 / 3 = 0.1667
I'm not confident with how I determined my sample space or event. Any help on how to approach these steps logically would help.
Best Answer
The sample space are $\{1,\ldots,6\}^{18}$, we are interested in those of which exactly $3$ of them takes value $5$. Out of the $18$ positions, we have to pick $3$ of them to take value $5$.
$$\frac{\binom{18}{3}\cdot 1^3\cdot 5^{15}}{6^{18}}$$
Relevant keyword: Binomial distribution.
Remark about your attempt:
$\{1,\ldots,6\}^{18}$ has cardinality $6^{18}$ rather than $6(18)$.