Question

In an examination $10$ questions of true false type are asked A student tosses a fair coin to determine his answer to each question if the coin falls heads, he answers true and if it false tails, he answers false. show that the probability that he answers at most $7$ questions correctly is $\frac{121}{128}$ probability

Answer 1

P(True)=P(head)$=\frac{1}{2}$

P(False)=P(tail)$=\frac{1}{2}$

P(7 questions correct)$={{}^{10}C}_7{\left(\frac{1}{2}\right)}^7{\left(\frac{1}{2}\right)}^3+{{}^{10}C}_6{\left(\frac{1}{2}\right)}^6{\left(\frac{1}{2}\right)}^4+.....{{}^{10}C}_0{\left(\frac{1}{2}\right)}^0{\left(\frac{1}{2}\right)}^{10}$

$=[{{}^{10}C}_0+{{}^{10}C}_1+{{}^{10}C}_2+......{{}^{10}C}_7]{\left(\frac{1}{2}\right)}^{10}=\frac{121}{128}$