$A$ student appeared in an examination consisting of $8$ true-false type questions. The student guesses the answers with equal probability. The smallest value of $n$,so that the probability of guessing at least $n$ correct answers is less than $\frac{1}{2}$,is:

If a Bernoulli trial is conducted $n$ times,then which one of the following is not suitable to use Poisson distribution?
$(i)$ Each trial results in two mutually exclusive outcomes namely success,failure.
(ii) The number $n$ of such trials is sufficiently large.
(iii) The trials are independent of each other.
(iv) The probability $p$ of success in each trial is very large.

Six balls are drawn successively from an urn containing $7$ red and $9$ black balls. Determine whether or not the trials of drawing balls are Bernoulli trials when the ball drawn is not replaced in the urn after each draw.

$A$ die is thrown $10$ times. The probability that an odd number will come up at least once is

The probability that a bulb produced by a factory will fuse after $150$ days of use is $0.05$. What is the probability that out of $5$ such bulbs,none will fuse after $150$ days of use?

If $\mu$ and $\sigma^{2}$ are the mean and variance of a random variable $X$ whose probability mass function is given by $P(X=x) = \binom{6}{x} \left(\frac{1}{3}\right)^{x} \left(\frac{2}{3}\right)^{6-x}$ for $x = 0, 1, 2, \ldots, 6$,then the value of $2\mu + 12\sigma^{2}$ is: