Theoretical DistributionsMCQMTP May 18Question 3433 of 230
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If mean and standard deviation of a binomial distribution is $10and\displaystyle and4respectively,\displaystyle respectively,p$ will be

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option d — $0.16$

All Options:

  • A$0.4$
  • B$0.44$
  • C$40$
  • D$0.16$

Detailed Solution & Explanation

**Binomial Distribution: Finding p from Mean and SD**\n\nGiven: Mean =np=10\displaystyle = np = 10, Standard Deviation =sqrtnpq=4\displaystyle = \npq=4\displaystyle \sqrt{npq} = 4\n\n**Step 1: Find the variance**\ntextVariance=(textSD)2=42=16\Variance\displaystyle \text{Variance} = (\SD\displaystyle \text{SD})^2 = 4^2 = 16\n\n**Step 2: Check validity**\nq=fractextVariancetextMean=frac1610=1.6q = \textVariancetextMean\displaystyle \frac{\\text{Variance}}{\\text{Mean}} = \1610=1\displaystyle \frac{16}{10} = 1.6\n\nSince q=1.6>1\displaystyle q = 1.6 > 1, this is **mathematically impossible** for a true binomial distribution (as q\displaystyle q must satisfy $0 < q < 1).\n\n∗∗Step3:Textbookapproach(asexpectedinexam)∗∗\n\nSometextbookscompute\displaystyle ).\n\n**Step 3: Textbook approach (as expected in exam)**\n\nSome textbooks computepusingtheratioformula:\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>p</mi><mo>=</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><mspacelinebreak="newline"></mspace><mi>t</mi><mi>e</mi><mi>x</mi><mi>t</mi><mrow><mi>V</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow></mrow><mrow><mostretchy="false">(</mo><mspacelinebreak="newline"></mspace><mi>t</mi><mi>e</mi><mi>x</mi><mi>t</mi><mrow><mi>M</mi><mi>e</mi><mi>a</mi><mi>n</mi></mrow><msup><mostretchy="false">)</mo><mn>2</mn></msup></mrow><mo>=</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>16</mn><mrow><mostretchy="false">(</mo><mn>10</mn><msup><mostretchy="false">)</mo><mn>2</mn></msup></mrow><mo>=</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>16100</mn><mo>=</mo><mn>0.16</mn></mrow><annotationencoding="application/x−tex">p=fractextVariance(textMean)2=frac16(10)2=frac16100=0.16</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal">p</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mspacenewline"></span><spanclass="mordmathnormal">t</span><spanclass="mordmathnormal">e</span><spanclass="mordmathnormal">x</span><spanclass="mordmathnormal">t</span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.22222em;">V</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">ian</span><spanclass="mordmathnormal">ce</span></span></span><spanclass="mord"><spanclass="mopen">(</span><spanclass="mspacenewline"></span><spanclass="mordmathnormal">t</span><spanclass="mordmathnormal">e</span><spanclass="mordmathnormal">x</span><spanclass="mordmathnormal">t</span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.10903em;">M</span><spanclass="mordmathnormal">e</span><spanclass="mordmathnormal">an</span></span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">16</span></span><spanclass="mord"><spanclass="mopen">(</span><spanclass="mord">10</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">16</span></span><spanclass="mord"><spanclass="mord">100</span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">0.16</span></span></span></span></span>\n\n>∗∗Note:∗∗Thedistributionwithmean=10,SD=4istechnically∗∗NotValid∗∗(asconfirmedinarelatedquestion).However,thetextbookanswerkeylists\displaystyle using the ratio formula:\n<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo>=</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><mspace linebreak="newline"></mspace><mi>t</mi><mi>e</mi><mi>x</mi><mi>t</mi><mrow><mi>V</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow></mrow><mrow><mo stretchy="false">(</mo><mspace linebreak="newline"></mspace><mi>t</mi><mi>e</mi><mi>x</mi><mi>t</mi><mrow><mi>M</mi><mi>e</mi><mi>a</mi><mi>n</mi></mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mo>=</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>16</mn><mrow><mo stretchy="false">(</mo><mn>10</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mo>=</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>16100</mn><mo>=</mo><mn>0.16</mn></mrow><annotation encoding="application/x-tex">p = \textVariance(textMean)2\displaystyle \frac{\\text{Variance}}{(\\text{Mean})^2} = \16(10)2\displaystyle \frac{16}{(10)^2} = \16100=0\displaystyle \frac{16}{100} = 0.16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord"><span class="mspace newline"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">e</span><span class="mord mathnormal">x</span><span class="mord mathnormal">t</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">ian</span><span class="mord mathnormal">ce</span></span></span><span class="mord"><span class="mopen">(</span><span class="mspace newline"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">e</span><span class="mord mathnormal">x</span><span class="mord mathnormal">t</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal">e</span><span class="mord mathnormal">an</span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord">16</span></span><span class="mord"><span class="mopen">(</span><span class="mord">10</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord">16</span></span><span class="mord"><span class="mord">100</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.16</span></span></span></span></span>\n\n> **Note:** The distribution with mean = 10, SD = 4 is technically **Not Valid** (as confirmed in a related question). However, the textbook answer key listsp = 0.16$ for this question.\n\nHence, **Option D** 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

Standard Deviation

A measure of the dispersion or spread of values in a data set around the mean. It is the square root of Variance. A lower standard deviation indicates data points are clustered close to the mean; a higher value indicates greater spread.

More Questions from Theoretical Distributions

Find the variance of binomial distribution with n=10\displaystyle n=10, p=0.3\displaystyle p=0.3

When p=0.5\displaystyle p=0.5, the binomial distribution is

If mean and standard deviation of a binomial distribution is $10and\displaystyle and2respectively,\displaystyle respectively,q$ will be

A random variable X\displaystyle X follows Binomial Distribution With E(x)=2\displaystyle E(x)=2 and V(x)=1.2\displaystyle V(x)=1.2, then the value of n\displaystyle n is

When 'p\displaystyle p' is large than $0.5$, the Binomial Distribution is:

For a Poisson distribution,

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