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

You roll the die 10 times and get exactly four 2's.

To compute the probability of rolling exactly four 2's in 10 rolls of a standard 4-sided die, we'll use the binomial probability formula.

### Steps:

1. **Identify the parameters:**
   - Number of trials (\(n\)): 10
   - Number of successes (\(k\)): 4 (rolling a 2 is considered a "success")
   - Probability of success on a single trial (\(p\)): \(\frac{1}{4} = 0.25\)
   - Probability of failure on a single trial (\(1 - p\)): \(\frac{3}{4} = 0.75\)

2. **Use the binomial probability formula:**

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

where:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

3. **Substitute the values:**

\[
P(X = 4) = \binom{10}{4} \times (0.25)^4 \times (0.75)^6
\]

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

4. **Calculate the probability:**

\[
P(X = 4) = 210 \times (0.25)^4 \times (0.75)^6
\]

\[
P(X = 4) = 210 \times (0.00390625) \times (0.177978515625)
\]

\[
P(X = 4) = 210 \times 0.00069427490234375
\]

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

So, the probability of rolling exactly four 2's in 10 rolls of a 4-sided die is **0.146** (rounded to three decimal places).

Would you like any more details or have any questions?

Here are five related questions:
1. What is the probability of getting exactly five 2's in 10 rolls of a 4-sided die?
2. How would the probability change if you wanted to roll exactly three 2's instead of four?
3. What is the probability of getting no 2's 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 1's in 10 rolls of a 4-sided die?

**Tip:** The binomial distribution is very useful when dealing with independent trials, like rolling a die multiple times, where each outcome is either a "success" or "failure".

Learn how to calculate the probability of rolling exactly four 2's in 10 rolls of a standard 4-sided die using the binomial probability formula. Get a step-by-step solution and understand the concept of binomial distribution in probability theory. Suitable for high school students studying probability.

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