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Binomial Distribution — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinomial Distribution3 Marks
Question
In a 20-question true-false examination, suppose a student tosses a fair coin to determine his answer to each question. For every head, he answers 'true' and for every tail, he answers 'false'. Find the probability that he answers at least 12 questions correctly.
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Answer

Let p be the probability of answering a true. So
$\text{p}=\frac{1}{2}$
$\text{q}=1-\frac{1}{2}$ [Since p +q = 1]
$=\frac{1}{2}$
Thus the probability that he answers at least 12 questions correctly among 20 questions is
$\text{P(X}\geq12)=\text{P(X}=12)+\text{P(X}=13)+\text{P(X}=14)+\text{P(X}=15)\\+\text{P(X}=16)+\text{P(X}=17)+\text{P(X}=18)+\text{P(X}=19)+\text{P(X}=20)$
$$$=\big(\frac{1}{2}\big)^{20}\big\{\text{ }^{20}\text{C}_{12}+\text{ }^{20}\text{C}_{13}+\text{ }^{20}\text{C}_{14}+\text{ }^{20}\text{C}_{15}\\+\text{ }^{20}\text{C}_{16}+\text{ }^{20}\text{C}_{17}+\text{ }^{20}\text{C}_{18}+\text{ }^{20}\text{C}_{19}+\text{ }^{20}\text{C}_{20}\big\}$
$\frac{\text{ }^{20}\text{C}_{12}+\text{ }^{20}\text{C}_{13}+\text{ }^{20}\text{C}_{14}+\text{ }^{20}\text{C}_{15}+\text{ }^{20}\text{C}_{16}+\text{ }^{20}\text{C}_{17}+\text{ }^{20}\text{C}_{18}+\text{ }^{20}\text{C}_{19}+\text{ }^{20}\text{C}_{20}}{2^{20}}$
Therefore, the required answer is
$\frac{\text{ }^{20}\text{C}_{12}+\text{ }^{20}\text{C}_{13}+\text{ }^{20}\text{C}_{14}+\text{ }^{20}\text{C}_{15}+\text{ }^{20}\text{C}_{16}+\text{ }^{20}\text{C}_{17}+\text{ }^{20}\text{C}_{18}+\text{ }^{20}\text{C}_{19}+\text{ }^{20}\text{C}_{20}}{2^{20}}$

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