We roll a fair 6-sided die 5 times. What is the probability that we get a 6 in at most 2 of the rolls?

Amanita, let's break down the problem step by step:

Step 1: Understand the problem

We roll a fair 6-sided die 5 times. We want the probability of getting a 6 in at most 2 of those rolls. "At most 2" means we can get a 6 zero times, one time, or two times across the 5 rolls.

Step 2: Choose the proper probability model

This scenario can be described using the binomial probability formula, because each roll is independent and there is only "success" (getting a 6) or "failure" (getting anything else) in each roll.

The binomial probability formula is:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

where:

• $n$ = total number of trials (rolls) = 5
• $k$ = number of desired successes (times we get a 6) = 0, 1, or 2
• $p$ = probability of success on a single roll = $\frac{1}{6}$

Step 3: Write the probability expression for "at most 2 sixes"

We need:

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$

Step 4: Calculate each term using the binomial formula

Let's substitute the values for each case:

• For $k=0$: $P(X=0) = \binom{5}{0} (\frac{1}{6})^0 (\frac{5}{6})^5$
• For $k=1$: $P(X=1) = \binom{5}{1} (\frac{1}{6})^1 (\frac{5}{6})^4$
• For $k=2$: $P(X=2) = \binom{5}{2} (\frac{1}{6})^2 (\frac{5}{6})^3$

Step 5: Sum all the probabilities to find the answer

Now, I will calculate and sum the values for $P(X=0)$, $P(X=1)$, and $P(X=2)$. Amanita, here is how we solve the problem step by step:

1. We use the binomial probability formula to find the probability of getting a 6 in at most 2 rolls when we roll a die 5 times.
2. We calculate the probability for getting 0, 1, or 2 sixes, and add them up.
3. The formula used is:

$P(X = k) = \binom{5}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\right)^{5-k}$

for $k = 0, 1, 2$. 4. Adding these probabilities, we get:

$P(X \leq 2) = P(X=0) + P(X=1) + P(X=2) = \frac{3125}{7776} + \frac{3125}{7776} + \frac{625}{3888} = \frac{625}{648}$

Final Answer:
The probability that you get a 6 in at most 2 of the 5 rolls is