To calculate the probability that an exam paper selected at random will be a failing paper, we need to know the failure threshold. Typically, a failing grade is defined as a score below 50 (assuming a 0–100 scale). However, if the failure threshold is different, please let me know.

Given:

• Mean score $\mu = 60$
• Standard deviation $\sigma = 10$
• Number of students = 40 (not directly needed for the probability calculation, but provides context)
• We will assume the failing grade is below 50.

Step 1: Convert the score to a z-score

The z-score represents how many standard deviations a value is from the mean. It is calculated by the formula:

$z = \frac{x - \mu}{\sigma}$

where:

• $x$ is the score in question (in this case, 50),
• $\mu$ is the mean (60),
• $\sigma$ is the standard deviation (10).

Substituting the values:

$z = \frac{50 - 60}{10} = \frac{-10}{10} = -1$

Step 2: Use the z-score to find the probability

Now that we have the z-score, we can use a standard normal distribution table (or a calculator) to find the probability that a score is below 50.

The z-score of -1 corresponds to a cumulative probability of approximately 0.1587. This means that about 15.87% of the students scored below 50, or the probability that a randomly selected exam paper is a failing one is 0.1587.

Final Answer:

The probability that the exam paper will be a failing paper is approximately 0.159 (rounded to three decimal places).

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Would you like further details or assistance with related concepts? Here are some follow-up questions:

1. How would the probability change if the failing grade was set at 55?
2. What would happen to the probability if the standard deviation were larger (say 15)?
3. How does the concept of the z-score relate to the normal distribution curve?
4. What is the role of the number of students (40) in calculating probabilities in this context?
5. How do you interpret a z-score of 0? What would that mean in this case?

Tip: Z-scores allow you to compare scores from different distributions. A z-score helps determine where a particular value lies relative to the rest of the data.