Let's calculate each probability step by step. The situation involves binomial probability, as each question has two possible outcomes (correct or incorrect), and the probability of success (correct answer) is constant for each trial.

Notations:

• $n = 10$: Total number of questions.
• $p = 0.25$: Probability of answering a question correctly.
• $q = 1 - p = 0.75$: Probability of answering a question incorrectly.
• $X$: Number of correct answers.

The binomial probability formula is:

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

where $\binom{n}{k}$ is the number of combinations of $n$ items taken $k$ at a time.

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(a) Probability that none of her answers are correct ($X = 0$):

$P(X = 0) = \binom{10}{0} (0.25)^0 (0.75)^{10}$

$P(X = 0) = 1 \cdot (0.75)^{10}$

$P(X = 0) = 0.0563 \quad (\text{approximately})$

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(b) Probability that she fails the exam (less than 5 correct answers):

Failing means $X < 5$, i.e., $P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$. Calculate each term:

$P(X = k) = \binom{10}{k} (0.25)^k (0.75)^{10-k}$

For each value of $k$:

• $P(X = 0) = 0.0563$ (from part a)
• $P(X = 1) = \binom{10}{1} (0.25)^1 (0.75)^9 = 10 \cdot 0.25 \cdot 0.1937 = 0.484$
• $P(X = 2) = \binom{10}{2} (0.25)^2 (0.75)^8 = 45 \cdot 0.0625 \cdot 0.256 = 0.2816$
• $P(X = 3) = \binom{10}{3} (0.25)^3 (0.75)^7 = 120 \cdot 0.015625 \cdot 0.3206 = 0.1503$
• $P(X = 4) = \binom{10}{4} (0.25)^4 (0.75)^6 = 210 \cdot 0.00390625 \cdot 0.4096 = 0.0401$

Summing these:

$P(\text{fail}) = 0.0563 + 0.484 + 0.2816 + 0.1503 + 0.0401 = 0.9863 \quad (\text{approximately})$

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(c) Probability that she passes the exam (5 or more correct answers):

Passing means $X \geq 5$. Since $P(X \geq 5) + P(X < 5) = 1$:

$P(\text{pass}) = 1 - P(\text{fail})$

$P(\text{pass}) = 1 - 0.9863 = 0.0137 \quad (\text{approximately})$

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Final Results:

• (a) Probability none of her answers are correct: $\mathbf{0.0563}$
• (b) Probability she fails the exam: $\mathbf{0.9863}$
• (c) Probability she passes the exam: $\mathbf{0.0137}$

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Do you want a deeper explanation or any further clarifications? Here are 5 related questions:

1. How does the probability change if the number of questions increases to 20?
2. What is the probability of getting exactly 5 correct answers?
3. How does the assumption of guessing affect these probabilities?
4. Can we plot the binomial distribution for $n = 10$ and $p = 0.25$?
5. What are the odds of passing if $p$ increases to 0.5?

Tip: The binomial distribution is ideal for scenarios where outcomes are independent and probabilities are fixed.