The experiment to to check if 5 bulbs drawn at random from a sample will fuse or not. It is a case of Bernoulli trials as it satisfies the conditions (i) finite number of trials, (ii) independent trials, (iii) there is a definite outcome and (iv) the probability of success does not change for each trial.

P (that a bulb will fuse after 150 days) = p = 0.05.

P (that a bulb won't fuse after 150 days) = q = 1- p = 0.95

If X is the number of bulbs that will fuse after 150 days in 5 trials, X has a bionomial distribution, the probability of x success in n-Bernoulli trials, $P (X = x) = \large^{n}C_x. p^x.q^{n–x}$ where $x = 0, 1, 2,...,n$ and $(q = 1 – p)$

Here n = 5, p = 0.05 and q = 0.95

(i) $P (X = 0) = \large^{5}C_0$$ \times (0.05)^0 \times (0.95)^5 = 0.95^5$

(ii) $P (X \leq 1 ) = P(X=0) + P (X=1)$

$P (X \leq 1) = 0.95^5+ \large^{5}C_1$$ \times (0.05)^1 \times (0.95)^(5-1) = 0.95^5+0.95^4(5\times0.05)$$

$P (X \leq 1) = 0.95^4 (0.95+0.25) = 1.2 \times 0.95^4$

(iii) $P(X\gt1) = 1 - P (X\leq1) = 1 - 1.2 \times 0.95^4$

(iv) $P (X \geq 1) = 1 - P (X = 0) = 1 - 0.95^5$