Sample 2026-27 Class 12 Maths Q16 Solution: Probability that it rains on three consecutive days

The question

The probability that it rains on a particular day is $\frac{1}{5}$ ⁠. Assuming weather conditions of any two days are independent, find the probability that it rains on three consecutive days.

(A) $\frac{4}{5}$

(B) $\frac{1}{5}$

(C) $\frac{1}{25}$

(D) $\frac{1}{125}$

Exam answer

Let $E_1, E_2, E_3$ be the events that it rains on day 1, day 2 and day 3, so $P(E_1) = P(E_2) = P(E_3) = \frac{1}{5}$ ⁠.

The days are independent, so

\begin{aligned} &P(E_1 \cap E_2 \cap E_3) \\ &= P(E_1)\,P(E_2)\,P(E_3) \end{aligned}

Why?

Rain on three consecutive days means rain on day 1 and day 2 and day 3, so we need the probability that all three events happen.

For independent events this is the product of their probabilities. That is the multiplication rule:

\begin{aligned} &P(E \cap F \cap G) \\ &= P(E)P(F)P(G) \end{aligned}

\begin{aligned} P(\text{rain on 3 days}) &= \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \\ &= \frac{1}{125} \end{aligned}

Why?

$5 \times 5 \times 5 = 125$ ⁠.

\begin{aligned} \frac{1}{5} \times \frac{1}{5} &= \frac{1 \times 1}{5 \times 5} = \frac{1}{25} \\ \frac{1}{25} \times \frac{1}{5} &= \frac{1 \times 1}{25 \times 5} = \frac{1}{125} \end{aligned}

Watch out: $\frac{1}{25}$ in (C) is rain on two days only, and $\frac{4}{5}$ in (A) is no rain on one day.

Answer: (D) $\dfrac{1}{125}$

Where this comes from

Chapter 13, Probability: Independent events: multiplication rule.

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Q16: Probability that it rains on three consecutive days