Probability & Statistics

Practise Probability & Statistics for FBISE Grade 9–10 Mathematics (Grade 9–10): 8 multiple-choice questions with instant checking and worked explanations. Free, no sign-up.

Tap an answer to check it · 8 questions

1. Question 1

   If each observation in a dataset with a mean of 15 and a standard deviation of 4 is multiplied by 3 and then increased by 5, what is the new variance of the dataset?
   • A12
   • B53
   • C48
   • D144
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   Answer: D) 144

   Scaling data by a factor k multiplies the variance by k²; here k = 3, so variance becomes 9·16 = 144. Adding a constant shifts the data but leaves variance unchanged.

2. Question 2

   A grouped frequency distribution with continuous class boundaries has a total frequency of 40. The cumulative frequency of the class preceding the median class is 15, the median class frequency is 10, and the class interval length is 5. If the lower class boundary $L$ of the median class satisfies $2L-10=50$, what is the median of the distribution?
   • A35.0
   • B30.5
   • C32.5
   • D27.5
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   Answer: C) 32.5

   First find the lower boundary L from the given equation, then apply the grouped‑data median formula using N/2, the cumulative frequency before the median class, the frequency of the median class, and the class width.

3. Question 3

   A fair $6$-sided die is rolled three times. Assuming each roll is independent, what is the probability that the arithmetic mean of the three results exceeds $4$?
   • A3/8
   • B5/27
   • C1/3
   • D7/27
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   Answer: D) 7/27

   To find the probability that the arithmetic mean of three rolls of a fair $6$-sided die exceeds $4$, first observe that the total sum of the three rolls must strictly exceed $4\times 3=12$, meaning the sum must be at least $13$. The total number of equally likely outcomes when rolling three $6$-sided dice is $6^3=216$. By symmetry or direct calculation, the number of outcomes yielding a sum of $13$ or greater is $56$ (out of $216$). Therefore, the required probability is $\frac{56}{216}=\frac{7}{27}$.

4. Question 4

   A dataset of 5 observations has a mean of 6 and a variance of 4. Later, it was discovered that one observation recorded as 10 was incorrect and should be replaced by the positive root of the quadratic equation $x^2-5x-6=0$. What is the correct standard deviation of the modified dataset?
   • A1.2
   • B0.16
   • C2.0
   • D0.4
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   Answer: D) 0.4

   Replacing the erroneous value 10 with the positive root 6 changes the data set. Re‑computing the variance with the new sum and sum of squares gives 0.16, whose square root is 0.4, the correct standard deviation.

5. Question 5

   What is the sum of probabilities of all possible outcomes in a sample space?
   • A1
   • B0.5
   • C0
   • D100
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   Answer: A) 1

   By definition, the probabilities of all mutually exclusive outcomes in a sample space must add up to $1$. This is a fundamental property of probability models. Therefore the correct answer is $1$. The option $0$ is incorrect because the total probability cannot be zero; $0.5$ is incorrect because a complete sample space always sums to $1$, not $0.5$; and $100$ is incorrect because probabilities are expressed as numbers between $0$ and $1$, not as percentages.

6. Question 6

   Which measure of central tendency can be determined graphically using the point of intersection of less-than and more-than cumulative frequency curves?
   • AMedian
   • BArithmetic mean
   • CGeometric mean
   • DMode
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   Answer: A) Median

   The point where the less‑than and more‑than cumulative frequency curves intersect marks the median because at that value half the observations lie below and half lie above. This property does not hold for the mean, geometric mean, or mode, which are found by other methods.

7. Question 7

   A fair six-sided die is rolled twice; what is the probability that the sum of the two numbers shown is a prime number greater than 7?
   • A$\frac{7}{36}$
   • B$\frac{1}{18}$
   • C$\frac{1}{12}$
   • D$\frac{5}{36}$
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   Answer: B) $\frac{1}{18}$

   Only the sum 11 is a prime number greater than 7. There are exactly two ordered outcomes that give this sum, (5,6) and (6,5). With 36 equally likely outcomes when rolling two dice, the probability is 2/36 = 1/18. The other options arise from common counting or simplification errors.

8. Question 8

   Why is the median often preferred over the arithmetic mean to represent the central tendency of income data in a city?
   • AIt can be calculated without ordering the dataset
   • BIt is not affected by extreme values or outliers
   • CIt incorporates every value in the dataset equally
   • DIt always equals the mode of the distribution
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   Answer: B) It is not affected by extreme values or outliers

   The median depends only on the position of the middle value after the data are ordered, so extreme high or low incomes do not change it. The mean, however, adds all values together, so a few very large or very small incomes can shift it dramatically. Hence the median is preferred for skewed income data.

More FBISE Grade 9–10 Mathematics topics

• Coordinate Geometry (3)
• Linear Equations & Inequalities (6)
• Quadratic Equations (5)
• Real & Complex Numbers (7)
• Sets, Functions & Groups (8)
• Trigonometry (6)