Question

The probability that it rains today is 0.4. If it rains today, the probability that it will rain tomorrow is 0.8. If it does not rain today, the probability that it will rain tomorrow is 0.7. If

P₁: denotes the probability that it does not rain today.

P₂: denotes the probability that it will not rain tomorrow if it rains today.

P₃: denotes the probability that it will rain tomorrow if it does not rain today.

P₄: denotes the probability that it will not rain tomorrow if it does not rain today.

1. Find the value of P₁ × P₄ − P₂ × P₃.
2. Calculate the probability of it raining tomorrow.

Answer 1

Since the event of raining today and not raining today are complementary events, if the probability that it rains today is 0.4, then the probability that it does not rain today is 1 − 0.4 = 0.6 ⇒ P₁ = 0.6.

If it rains today, the probability that it will rain tomorrow is 0.8, and the probability that it will not rain tomorrow is 1 − 0.8 = 0.2.

If it does not rain today, the probability that it will rain tomorrow is 0.7, then the probability that it will not rain tomorrow is 1 − 0.7 = 0.3.

i. P₁ × P₄ − P₂ × P₃ = 0.6 × 0.3 − 0.2 × 0.7 = 0.0.4.

ii. Let E₁ and E₂ be the events that it will rain today and it will not rain today, respectively.

P(E₁) = 0.4 and P(E₂) = 0.6

A be the event that it will rain tomorrow.

`P (A/E_1)` = 0.8 and

`P (A/E_2)` = 0.7

We have,

`P(A) = P(E_1)P(A/E_1) + P(E_2)P(A/E_2)`

= 0.4 × 0.8 + 0.6 × 0.7

= 0.74