30% of students scored distinction in a test, if 10 students who took the test were chosen at random, find the probability that at least 2 of them get distinction.

The problem can be solved using the **binomial probability distribution**, as we are dealing with a fixed number of trials (10 students), two possible outcomes (distinction or no distinction), and a constant probability of success (30% or 0.3).

### Step 1: Identify the parameters
- \( n = 10 \) (number of students selected)
- \( p = 0.30 \) (probability of a student scoring a distinction)
- \( q = 1 - p = 0.70 \) (probability of a student not scoring a distinction)
- We are asked to find the probability that **at least 2** students get distinction, i.e., \( P(X \geq 2) \).

### Step 2: Set up the binomial probability formula
The general binomial probability formula is:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient, or "n choose k",
- \( p^k \) is the probability of \( k \) successes,
- \( q^{n-k} \) is the probability of \( n - k \) failures.

### Step 3: Find \( P(X \geq 2) \)
To find the probability that at least 2 students get a distinction, we can use the complement rule:
\[
P(X \geq 2) = 1 - P(X < 2) = 1 - [P(X = 0) + P(X = 1)]
\]

#### Step 4: Calculate \( P(X = 0) \) and \( P(X = 1) \)
1. **For \( X = 0 \)** (no students score distinction):
\[
P(X = 0) = \binom{10}{0} \cdot (0.30)^0 \cdot (0.70)^{10} = 1 \cdot 1 \cdot (0.70)^{10} = (0.70)^{10} \approx 0.0282
\]

2. **For \( X = 1 \)** (one student scores distinction):
\[
P(X = 1) = \binom{10}{1} \cdot (0.30)^1 \cdot (0.70)^9 = 10 \cdot (0.30) \cdot (0.70)^9
\]
\[
P(X = 1) \approx 10 \cdot 0.30 \cdot 0.0403 = 0.121
\]

### Step 5: Calculate \( P(X \geq 2) \)
Now, using the complement rule:
\[
P(X \geq 2) = 1 - [P(X = 0) + P(X = 1)] = 1 - [0.0282 + 0.121] = 1 - 0.1492 = 0.8508
\]

Thus, the probability that at least 2 of the 10 students get a distinction is approximately **0.8508** or **85.08%**.

Would you like any further details?

### Related Questions:
1. How would the probability change if only 5 students were chosen?
2. What is the probability that exactly 3 students scored distinction?
3. Can the binomial distribution be approximated using the normal distribution for large \(n\)?
4. How would this scenario change if the probability of distinction was 50%?
5. What is the expected number of students scoring distinction out of 10?

**Tip:** When dealing with "at least" or "at most" probabilities, always consider using the complement rule to simplify calculations.

Learn how to calculate the probability that at least 2 out of 10 students score a distinction in a test where the probability of distinction is 30%. Using binomial distribution and the complement rule, this detailed solution is suitable for Grades 10-12 students.

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