Mathematics - MCQ Practice Questions

Understanding:

We need the probability that both balls drawn (without replacement) are of the same colour.

Formula:

Step 1: Compute the total ways to choose 2 balls from 12.

Step 2: Compute favourable outcomes (both red, both blue, or both green).

Step 3: Add favourable outcomes.

Step 4: Compute the probability.

Answer:

The probability that both balls are of the same colour is $\frac{19}{66}$.

Quick Tip:

For "same colour" problems without replacement, always use combinations $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ per colour — never multiply probabilities sequentially unless you adjust the denominator at each draw.

Understanding:

We are told the mean of 10 observations is 15, and one value is replaced.

Formula:

Step 1: Find the original sum.

Step 2: Find the new sum after replacement.

Step 3: Compute the new mean.

Answer:

The new mean after replacing 20 with 30 is 16.

Quick Tip:

Whenever a single value is replaced, the mean changes by $\frac{\text{(new value)}-\text{(old value)}}{n}$. Here: $\frac{30-20}{10}=1$, so new mean $=15+1=16$.

Understanding:

Two fair dice are thrown. We need the probability that their sum is prime.

Formula:

Step 1: Identify prime sums in range $[2,12]$: $2,3,5,7,11$.

Step 2: Count outcomes for each prime sum.

Step 3: Total favourable outcomes.

Step 4: Compute the probability.

Answer:

The probability that the sum is a prime number is $\frac{5}{12}$.

Quick Tip:

Note that $4$ is not prime ($4=2\times 2$), and neither are $6,8,9,10,12$. A common error is to include $4$ or miss $11$.

Understanding:

We need to find the variance of the data set $\{2,4,6,8,10\}$.

Formula:

Step 1: Compute the mean.

Step 2: Compute each squared deviation.

Step 3: Sum the squared deviations.

Step 4: Divide by $n$.

Answer:

The variance of the given data set is $8$.

Quick Tip:

For an arithmetic progression $a,a+d,a+2d,\dots$ with $n$ terms, variance $=\frac{(n^2-1)d^2}{12}$. Here $d=2$, $n=5$: $\frac{(25-1)\times 4}{12}=\frac{96}{12}=8$. ✓

Understanding:

We need to find the conditional probability $P(A\mid B)$.

Formula:

Step 1: Find $P(A\cap B)$.

Step 2: Compute $P(A\mid B)$.

Answer:

The conditional probability $P(A\mid B)=0.4$.

Quick Tip:

A quick sanity check: since $P(A\mid B)=0.4=P(A)$, the events $A$ and $B$ are independent. This is consistent — independent events satisfy $P(A\cap B)=P(A)\cdot P(B)=0.4\times 0.5=0.2$. ✓

Understanding:

Using inclusion–exclusion to find students who passed in at least one subject, then subtracting from the total.

Formula:

Step 1: Find students who passed in at least one subject.

Step 2: Find students who failed in both.

Answer:

The number of students who failed in both subjects is $15$.

Understanding:

We need to compute the expected value (mean) of a discrete random variable $X$.

Formula:

Step 1: Multiply each value by its probability.

Step 2: Sum the products.

Verification:

Check probabilities sum to 1: $0.1+0.3+0.4+0.2=1.0$ ✓

Answer:

The expected value $E(X)=2.7$.

Quick Tip:

A common mistake is to simply average the $x$-values as $\frac{1+2+3+4}{4}=2.5$, ignoring the non-uniform probabilities.

Understanding:

We need to find the median of a grouped frequency distribution.

Formula:

where $L$ = lower boundary of median class, $C_f$ = cumulative frequency before median class, $f$ = frequency of median class, $h$ = class width.

Step 1: Find $\frac{N}{2}$.

Step 2: Build cumulative frequencies.

The cumulative frequency first exceeds 25 in the class $20\text{–}30$, so the median class is $20\text{–}30$.

Step 3: Identify the values.

Step 4: Apply the formula.

Answer:

The median of the distribution is $28$.

Quick Tip:

Always locate the median class by finding the class whose cumulative frequency first meets or exceeds $\frac{N}{2}$, not the class with the highest frequency.

Understanding:

We need the probability of drawing a face card or a spade from a standard 52-card deck.

Formula:

Step 1: Write each probability.

Step 2: Apply inclusion–exclusion.

Answer:

The probability of drawing a face card or a spade is $\frac{11}{26}$.

Quick Tip:

In a standard deck: 4 suits of 13 cards each, with 3 face cards per suit. Always subtract the overlap when using $P(A\cup B)$.

Understanding:

We need the standard deviation of the set $\{1,2,3,\dots ,n\}$.

Formula:

Step 1: Find the mean.

Step 2: Find $\frac{1}{n}\sum x_i^2$.

Step 3: Compute variance.

Step 4: Take the square root.

Answer:

The standard deviation of the first $n$ natural numbers is $\sqrt{\frac{n^2-1}{12}}$.

Quick Tip:

For $n=5$: $\sigma =\sqrt{\frac{24}{12}}=\sqrt{2}\approx 1.414$. You can verify this against $\{1,2,3,4,5\}$ (mean $=3$, variance $=\frac{4+1+0+1+4}{5}=2$). ✓

Understanding:

We need $P(\bar{A}\cap \bar{B})$, the probability that neither $A$ nor $B$ occurs.

Formula:

Step 1: Compute $P(A\cup B)$.

Step 2: Use LCM $=12$ to add fractions.

Step 3: Apply De Morgan's law.

Answer:

The probability that neither $A$ nor $B$ occurs is $\frac{5}{12}$.

Quick Tip:

De Morgan's law for events: $\bar{A}\cap \bar{B}=\overline{A\cup B}$. This converts the problem into finding $P(A\cup B)$ first, which is straightforward via inclusion–exclusion.