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 and solve it step by step!

Step 1: Understanding the Problem

We roll a fair 6-sided die 5 times. We want to find the probability that the number 6 appears in at most 2 of those rolls. This means that 6 can appear 0, 1, or 2 times in the five rolls.

Step 2: Recognizing the Probability Model

Each roll can be considered a "success" (rolling a 6) or "failure" (not rolling a 6). The probability of rolling a 6 on any one roll is:

and the probability of not rolling a 6 is:

Given 5 rolls, and the event "rolling a 6" can happen 0, 1, or 2 times, this follows a binomial distribution with parameters $n = 5$ and $p = \frac{1}{6}$.

Step 3: Writing the General Formula

The probability of getting exactly $k$ sixes in 5 rolls is:

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

where $C(5, k)$ is the number of ways to choose $k$ successful rolls out of 5.

Step 4: Find the Probability for Each Case

We want $P(X \leq 2)$, which means we sum the probabilities for $k = 0, 1, 2$:

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

Let’s calculate each term.

Step 5: Substitute and Calculate Each Term

1. For $k = 0$:

$P(X = 0) = C(5, 0) \cdot \left(\frac{1}{6}\right)^0 \cdot \left(\frac{5}{6}\right)^5$

2. For $k = 1$:

$P(X = 1) = C(5, 1) \cdot \left(\frac{1}{6}\right)^1 \cdot \left(\frac{5}{6}\right)^4$

3. For $k = 2$:

$P(X = 2) = C(5, 2) \cdot \left(\frac{1}{6}\right)^2 \cdot \left(\frac{5}{6}\right)^3$

Now, I will calculate each probability and then find their sum.

Step 6: Sum the Probabilities

After finding each probability, we add them:

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

I’ll now calculate all needed values and provide the answer. Amanita, here is the final answer:

The probability that you get a 6 in at most 2 of the 5 rolls of a fair 6-sided die is

This means that if you roll a die 5 times, in most cases (about 50% of the time), you will get a 6 no more than 2 times.