In each true and false question, probability
of guessing correctly is, $p=\frac{1}{2}$ and probability of
not guessing correctly, $q=\frac{1}{2}$.
Here, $n=10$
$\therefore $ The probability of guessing atleast 7 correctly
$=P(X \geq 7)$
$=P(X=7)+P(X=8)+P(X=9)+P(X=10)$
$={ }^{10} C_{7}\left(\frac{1}{2}\right)^{7}\left(\frac{1}{2}\right)^{3}+{ }^{10} C_{8}\left(\frac{1}{2}\right)^{8}\left(\frac{1}{2}\right)^{2}$
$+{ }^{10} C_{9}\left(\frac{1}{2}\right)^{9}\left(\frac{1}{2}\right)^{1}+{ }^{10} C_{10}\left(\frac{1}{2}\right)^{10}$
$\left[\because P(x=r)={ }^{n} C_{r} p^{\prime} q^{n-r}\right]$
$=120 \times\left(\frac{1}{2}\right)^{10}+45\left(\frac{1}{2}\right)^{10}+10\left(\frac{1}{2}\right)^{10}+1\left(\frac{1}{2}\right)^{10}$
$=\frac{120+45+10+1}{2^{10}}=\frac{176}{1024}=\frac{11}{64}$