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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability3 Marks
Question
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
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Answer

The repeated guessing of correct answers from multiple choice questions are Bernoulli trials. Let X represent the number of correct answers by guessing in the set of 5 multiple choice questions.
Probability of getting a correct answer is, $\text{p}=\frac{1}{3}$
$\therefore\ \text{q}=1-\text{p}=1-\frac{1}{3}=\frac{2}{3}$
Clearly, X has a binomial distribution with n = 5 and $\text{p}=\frac{1}{3}$
$\therefore\ \text{P}(\text{X=x})=\ ^\text{n}\text{C}_\text{x}\text{q}^\text{n-x}\text{p}^\text{x}$
$=\ ^5\text{C}_\text{x}\bigg(\frac{2}{3}\bigg)^{5-\text{x}}.\bigg(\frac{1}{3}\bigg)^\text{x}$
P(guessing more than 4 correct answers) = P(X ≥ 4)
$=\text{P}(\text{X}=4)+\text{P}(\text{X}=5)$
$=\ ^5\text{C}_\text{4}\bigg(\frac{2}{3}\bigg).\bigg(\frac{1}{3}\bigg)^4+\ ^5\text{C}_\text{5}\bigg(\frac{1}{3}\bigg)^5$
$=5\cdot\frac{2}{3}\cdot\frac{1}{81}+1\cdot\frac{1}{243}$
$=\frac{10}{243}+\frac{1}{243}$
$=\frac{11}{243}$

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