We are dealing with a binomial probability distribution in this scenario. The binomial distribution applies here because:

• Each student either passes or fails the midterm test (two possible outcomes).
• The probability of passing is constant for each student.
• The results of one student's test do not affect the others (independence).

Let:

• $n = 15$ (the number of students in the tutorial class),
• $p = 0.90$ (the probability of passing),
• $q = 1 - p = 0.10$ (the probability of failing).

(i) Probability that all students pass the midterm test:

For all students to pass the test, all 15 students need to succeed. The probability of this happening is:

$P(\text{All pass}) = p^{15} = 0.90^{15}$

Let’s compute that.

$P(\text{All pass}) = (0.90)^{15} \approx 0.2059$

So, the probability that all students pass the test is approximately 0.2059 (or 20.59%).

(ii) Probability that more than two students failed the midterm test:

We are asked to find the probability that more than two students failed, i.e., $P(X > 2)$, where $X$ is the number of students who failed. To calculate this, it's easier to find the probability that two or fewer students failed and subtract it from 1:

$P(X > 2) = 1 - P(X \leq 2)$

Where $P(X \leq 2)$ represents the cumulative probability for 0, 1, or 2 students failing. We will use the binomial probability formula:

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

We need to calculate the probabilities for $k = 0$, $k = 1$, and $k = 2$, then sum them up.

1. For $X = 0$ (no students fail): $P(X = 0) = \binom{15}{0} \cdot (0.90)^{15} \cdot (0.10)^0 = 0.2059$

2. For $X = 1$ (1 student fails): $P(X = 1) = \binom{15}{1} \cdot (0.90)^{14} \cdot (0.10)^1 = 15 \cdot (0.90)^{14} \cdot 0.10$

3. For $X = 2$ (2 students fail): $P(X = 2) = \binom{15}{2} \cdot (0.90)^{13} \cdot (0.10)^2 = \frac{15 \cdot 14}{2} \cdot (0.90)^{13} \cdot 0.10^2$

After finding these probabilities, sum them and subtract from 1 to find $P(X > 2)$. Let me calculate that.### (i) The probability that all 15 students pass the midterm test is approximately 0.2059 (or 20.59%).

(ii) The probability that more than two students fail the midterm test is approximately 0.1841 (or 18.41%).

Would you like more details on any of the steps?

Here are five related questions:

1. What is a binomial distribution, and how is it applied in probability problems?
2. How would the result change if the probability of passing was lower, such as 80%?
3. How do we interpret the result of the binomial coefficient in real-world scenarios?
4. What are other methods for calculating probabilities involving multiple trials?
5. How can cumulative probabilities be useful in different statistical scenarios?

Tip: In binomial distributions, calculating cumulative probabilities for a range of outcomes can simplify finding more complex probabilities like "more than" or "less than" scenarios.