Compute the probability (rounded to 3 decimal places) of the following events related to rolling a standard 4-sided die (with faces labeled 1,2,3, and 4:)

You roll the die 10 times and get exactly four 22s

To solve this problem, we need to compute the probability of rolling exactly four 2s when rolling a standard 4-sided die 10 times.

The number of ways to choose exactly four 2s in 10 rolls is given by the binomial coefficient, denoted as \(\binom{n}{k}\), where \(n\) is the total number of rolls (10) and \(k\) is the number of times you want to roll a 2 (4).

The probability of rolling a 2 on a single roll of a 4-sided die is \(\frac{1}{4}\), and the probability of not rolling a 2 is \(\frac{3}{4}\).

The probability of exactly four 2s in 10 rolls is given by the binomial distribution formula:

\[
P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}
\]

where:
- \(n = 10\) (number of trials),
- \(k = 4\) (number of successful outcomes),
- \(p = \frac{1}{4}\) (probability of rolling a 2),
- \(1-p = \frac{3}{4}\) (probability of not rolling a 2).

Let's compute this step by step.

\[
\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

\[
P(X = 4) = 210 \times \left(\frac{1}{4}\right)^4 \times \left(\frac{3}{4}\right)^{6}
\]

\[
P(X = 4) = 210 \times \left(\frac{1}{256}\right) \times \left(\frac{729}{4096}\right)
\]

\[
P(X = 4) = 210 \times \frac{729}{1048576}
\]

\[
P(X = 4) \approx 0.146
\]

Thus, the probability of getting exactly four 2s in 10 rolls of a standard 4-sided die is approximately **0.146**.

Would you like further details on this calculation, or do you have any other questions?

Here are five related questions:
1. What is the probability of getting exactly three 2s in 10 rolls of a 4-sided die?
2. How would the probability change if you wanted to roll exactly four 3s instead?
3. What is the probability of getting at least one 2 in 10 rolls?
4. How does the probability change if you roll the die 12 times instead of 10 times?
5. What is the probability of getting exactly four 2s in 8 rolls of a 4-sided die?

**Tip:** When working with binomial probabilities, it's useful to familiarize yourself with the binomial distribution and the use of combinations (binomial coefficients).

Learn how to calculate the probability of rolling exactly four 2s in 10 rolls of a standard 4-sided die. Step-by-step solution using binomial distribution and binomial coefficient. Suitable for Grades 10-12.

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