Find mode when n = 15 and p = 1/4 in binomial distribution?

Option a: 4. Solution: Mode of Binomial Distribution: n = 15 , p = 1/4 Given: n = 15 , p = 1/4 Formula for Mode of Binomial Distribution: Compute (n+1)p : (n+1)p = (15+1) × 1/4 = 16 × 1/4 = 4 Rule: - If (n+1)p is an integer, then the distribution has two modes: (n+1)p and (n+1)p - 1 , i.e., 4 and 3. - If (n+1)p is not an integer, the unique mode is (n+1)p . Since (n+1)p = 4 is an integer, technically both 3 and 4 are modes. However, in standard CA Foundation textbook convention, when (n+1)p is exactly an integer, the mode is taken as (n+1)p = 4 (the larger of the two modal values). Some textbooks also write mode as np + p = 4 = 4 . Also note: The mean = np = 15 × 1/4 = 3.75 . The mode for a discrete distribution near 3.75 is typically 4 (the nearest integer rounding up when (n+1)p is exactly integer). The textbook answer is 4, accepting the larger modal value. Hence, Option A is the correct answer.

Find mode when $n = 15$ and $p = 1/4$ in binomial distribution?

The correct answer is Option a: $4$. **Mode of Binomial Distribution: $n = 15$, $p = 1/4$** **Given:** $n = 15$, $p = \\frac{1}{4}$ **Formula for Mode of Binomial Distribution:** Compute $(n+1)p$: $$(n+1)p = (15+1) \times \\frac{1}{4} = 16 \times \\frac{1}{4} = 4$$ **Rule:** - If $(n+1)p$ is an **integer**, then the distribution has **two modes**: $(n+1)p$ and $(n+1)p - 1$, i.e., **4 and 3**. - If $(n+1)p$ is **not an integer**, the unique mode is $\lfloor (n+1)p \rfloor$. Since $(n+1)p = 4$ is an integer, technically both **3 and 4** are modes. **However**, in standard CA Foundation textbook convention, when $(n+1)p$ is exactly an integer, the mode is taken as $(n+1)p = 4$ (the larger of the two modal values). Some textbooks also write mode as $\lfloor np + p \rfloor = \lfloor 4 \rfloor = 4$. **Also note:** The mean $= np = 15 \times \\frac{1}{4} = 3.75$. The mode for a discrete distribution near 3.75 is typically 4 (the nearest integer rounding up when (n+1)p is exactly integer). The textbook answer is **4**, accepting the larger modal value. Hence, **Option A** is the correct answer.

Theoretical DistributionsMCQPYQ Nov 19Question 3419 of 230

All Questions

Find mode when $\displaystyle n = 15$ and $\displaystyle p = 1/4$ in binomial distribution?

Options

\displaystyle 4

\displaystyle 4

and

\displaystyle 3

\displaystyle 4.2

\displaystyle 3.75

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option a — $\displaystyle 4$

All Options:

A $\displaystyle 4$
B $\displaystyle 4$ and $\displaystyle 3$
C $\displaystyle 4.2$
D $\displaystyle 3.75$

Detailed Solution & Explanation

**Mode of Binomial Distribution:

\displaystyle n = 15

\displaystyle p = 1/4

** **Given:**

\displaystyle n = 15

\displaystyle p = \\frac{1}{4}

**Formula for Mode of Binomial Distribution:** Compute

\displaystyle (n+1)p

(n+1)p = (15+1) \times \\frac{1}{4} = 16 \times \\frac{1}{4} = 4

**Rule:** - If

\displaystyle (n+1)p

is an **integer**, then the distribution has **two modes**:

\displaystyle (n+1)p

and

\displaystyle (n+1)p - 1

, i.e., **4 and 3**. - If

\displaystyle (n+1)p

is **not an integer**, the unique mode is

\displaystyle \lfloor (n+1)p \rfloor

. Since

\displaystyle (n+1)p = 4

is an integer, technically both **3 and 4** are modes. **However**, in standard CA Foundation textbook convention, when

\displaystyle (n+1)p

is exactly an integer, the mode is taken as

\displaystyle (n+1)p = 4

(the larger of the two modal values). Some textbooks also write mode as

\displaystyle \lfloor np + p \rfloor = \lfloor 4 \rfloor = 4

. **Also note:** The mean

\displaystyle = np = 15 \times \\frac{1}{4} = 3.75

. The mode for a discrete distribution near 3.75 is typically 4 (the nearest integer rounding up when (n+1)p is exactly integer). The textbook answer is **4**, accepting the larger modal value. Hence, **Option A** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Mode

The value that appears most frequently in a data set. A data set can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes). It is the only measure of central tendency applicable to qualitative (categorical) data.

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Find mode when n=15\displaystyle n = 15n=15 and p=1/4\displaystyle p = 1/4p=1/4 in binomial distribution?