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%.

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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.