The probability of finding a green signal on a busy crossing $X$ is 30%. What is the probability of finding a green signal on $X$ on two consecutive days out of three?

$0.126$

The correct answer is Option (1) → $0.126$ ##

Let $A$ be the event of finding a green signal on 2 consecutive days.

Let $B_i, i=1,2,3$ be the probability of finding a green signal on $i^{th}$ day.

$P(B_1) = P(B_2) = P(B_3) = \frac{30}{100}$

$ P(A) = P(B_1)P(B_2)P(\overline{B_3}) + P(\overline{B_1})P(B_2)P(B_3)$

$= \left( \frac{30}{100} \times \frac{30}{100} \times \frac{70}{100} \right) + \left( \frac{70}{100} \times \frac{30}{100} \times \frac{30}{100} \right) $

$= \frac{63000 + 63000}{1000000} = \frac{126000}{1000000} = \frac{126}{1000} \text{ or } 12.6\%$