15 extra multiple-choice questions for Statistics and Probability (10th Standard Mathematics, Samacheer Kalvi), beyond the ones printed in the textbook — each with the correct option highlighted and a clear, worked explanation. Free to read in English and Tamil.
Q1
If the largest value in a data set is \(85\) and its range is \(38\), what is the smallest value in the data set?
- A. \(47\)Correct
- B. \(123\)
- C. \(38\)
- D. \(57\)
Explanation. The range of a data set is the difference between the largest value and the smallest value, \(R = L - S\). Substituting \(R = 38\) and \(L = 85\), we get the smallest value as \(S = 85 - 38 = 47\).
Q2
If the standard deviation of a set of observations is \(4.2\) and each observation is increased by \(3.5\), what is the standard deviation of the new set of observations?
- A. \(7.7\)
- B. \(4.2\)Correct
- C. \(1.2\)
- D. \(14.7\)
Explanation. Adding a constant to each value in a data set does not change its standard deviation. Therefore, the standard deviation remains \(4.2\).
Q3
If the variance of a set of observations is \(9\), and each observation is multiplied by \(3\), what is the standard deviation of the new set of observations?
- A. \(9\)Correct
- B. \(27\)
- C. \(81\)
- D. \(3\)
Explanation. The original standard deviation is the square root of the variance, which is \(\sqrt{9} = 3\). Multiplying each value by a constant \(k\) multiplies the standard deviation by \(k\). Here, the new standard deviation is \(3 \times 3 = 9\).
Q4
Find the coefficient of variation of a data set if its mean is \(20\) and its standard deviation is \(5\).
- A. \(25\%\)Correct
- B. \(400\%\)
- C. \(15\%\)
- D. \(4\%\)
Explanation. The coefficient of variation is calculated as \(C.V. = \frac{\sigma}{\bar{x}} \times 100\%\). Substituting \(\sigma = 5\) and \(\bar{x} = 20\), we get \(C.V. = \frac{5}{20} \times 100\% = 25\%\).
Q5
Two groups A and B have coefficients of variation \(12.5\%\) and \(18.2\%\) respectively. Which of the following statements is true regarding their consistency?
- A. Group B is more consistent than Group A
- B. Group A is more consistent than Group BCorrect
- C. Both groups are equally consistent
- D. Consistency cannot be compared using coefficient of variation
Explanation. A lower coefficient of variation indicates greater consistency in the data. Since Group A has a lower coefficient of variation (\(12.5\%\)) compared to Group B (\(18.2\%\)), Group A is more consistent than Group B.
Q6
What is the sum of the probability of a sure event and the probability of an impossible event?
- A. \(0\)
- B. \(0.5\)
- C. \(1\)Correct
- D. \(2\)
Explanation. The probability of a sure event is always \(1\), and the probability of an impossible event is always \(0\). Their sum is \(1 + 0 = 1\).
Q7
If the probability of occurrence of an event \(E\) is \(0.35\), what is the probability of the complement event \(\overline{E}\)?
- A. \(0.35\)
- B. \(0.65\)Correct
- C. \(1.35\)
- D. \(-0.35\)
Explanation. The probability of the complement of an event is given by \(P(\overline{E}) = 1 - P(E)\). Substituting \(P(E) = 0.35\) into this formula gives the complement probability as \(1 - 0.35 = 0.65\).
Q8
Three unbiased coins are tossed simultaneously. What is the probability of getting at least two heads?
- A. \(\frac{1}{2}\)Correct
- B. \(\frac{3}{8}\)
- C. \(\frac{7}{8}\)
- D. \(\frac{1}{4}\)
Explanation. The sample space of tossing three coins has \(8\) equally likely outcomes. The outcomes with at least two heads are HHH, HHT, HTH, and THH. This gives \(4\) favorable outcomes, so the probability is \(\frac{4}{8} = \frac{1}{2}\).
Q9
For any two events \(A\) and \(B\), if \(P(A) = 0.5\), \(P(B) = 0.4\), and \(P(A \cup B) = 0.7\), what is \(P(A \cap B)\)?
- A. \(0.2\)Correct
- B. \(0.9\)
- C. \(0.1\)
- D. \(0.3\)
Explanation. Using the addition theorem of probability, we have \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Substituting the given values, \(0.7 = 0.5 + 0.4 - P(A \cap B)\), which simplifies to \(P(A \cap B) = 0.2\).
Q10
If \(A\) and \(B\) are two mutually exclusive events such that \(P(A) = 0.3\) and \(P(B) = 0.4\), what is \(P(A \cup B)\)?
- A. \(0.7\)Correct
- B. \(0.12\)
- C. \(0.1\)
- D. \(0.5\)
Explanation. For mutually exclusive events, the intersection probability \(P(A \cap B)\) is \(0\). Therefore, the addition theorem simplifies to \(P(A \cup B) = P(A) + P(B)\). This gives \(0.3 + 0.4 = 0.7\).
Q11
What is the variance of the first \(5\) natural numbers?
- A. \(2\)Correct
- B. \(3\)
- C. \(2.5\)
- D. \(1.5\)
Explanation. The variance of the first \(n\) natural numbers is calculated using the formula \(\frac{n^2 - 1}{12}\). For \(n = 5\), substituting the value yields a variance of \(\frac{5^2 - 1}{12} = \frac{24}{12} = 2\).
Q12
When two fair dice are rolled together, what is the total number of sample points in the sample space?
- A. \(12\)
- B. \(36\)Correct
- C. \(6\)
- D. \(18\)
Explanation. When two fair dice are rolled together, each die has \(6\) possible outcomes. By the fundamental counting principle, the total number of sample points in the sample space is \(6 \times 6 = 36\).
Q13
What is the probability that a non-leap year selected at random will contain \(53\) Sundays?
- A. \(\frac{2}{7}\)
- B. \(\frac{1}{7}\)Correct
- C. \(\frac{53}{365}\)
- D. \(\frac{1}{365}\)
Explanation. A non-leap year has \(365\) days, which equals \(52\) weeks and \(1\) remaining day. This remaining day can be any of the \(7\) days of the week. The probability of it being a Sunday is therefore \(\frac{1}{7}\).
Q14
Two dice are rolled simultaneously. What is the probability of getting a doublet (equal numbers on both dice)?
- A. \(\frac{1}{36}\)
- B. \(\frac{1}{6}\)Correct
- C. \(\frac{1}{2}\)
- D. \(\frac{5}{6}\)
Explanation. When two dice are rolled simultaneously, there are \(36\) total outcomes. The favorable outcomes for a doublet are \((1,1)\), \((2,2)\), \((3,3)\), \((4,4)\), \((5,5)\), and \((6,6)\) (total \(6\) outcomes). The probability is \(\frac{6}{36} = \frac{1}{6}\).
Q15
What is the range of the first \(5\) prime numbers?
- A. \(9\)Correct
- B. \(11\)
- C. \(7\)
- D. \(8\)
Explanation. The first \(5\) prime numbers are \(2, 3, 5, 7,\) and \(11\). The largest value \(L\) is \(11\) and the smallest value \(S\) is \(2\). The range is \(L - S = 11 - 2 = 9\).
