Question

A gardener wants to plant saplings on a day when rain is not predicted. According to the forecast by the weather department,

• the probability of rain today is 0.4.
• if it rains today, the probability of it raining tomorrow is 0-8.
• if it does not rain today, the probability that it will rain tomorrow is 0.7.

(i) What is the probability that he will not plant the saplings tomorrow?   [2]

(ii) Find the probability that he will plant them tomorrow.    [1]

(iii) Given that he does not plant them tomorrow, what is the probability that he did not plant them today?   [2]

(iv) What is the probability that he can plant saplings on both days?   [1]

Answer 1

Let R₁ be the event that it rains today.

Let R₂ be the event that it rains tomorrow.

Gardener’s Rule: He plants only when it does not rain.

“Planting” = No Rain.

“Not Planting” = Rain.

(i) “Not planting tomorrow” means it will rain tomorrow (R₂). We use the Law of Total Probability:

P(R₂) = [P(R₁) × P(R₂|R₁] + [P(R′₁) × P(R₂|R′₁)]

P(R₂) = (0.4 × 0.8) + (0.6 × 0.7)

P(R₂) = 0.32 + 0.42 = 0.74

(ii) “Planting tomorrow” means there is no rain tomorrow (R′2). This is the complement of the probability found in part (i):

P(Plant Tomorrow) = 1 − P(R₂)

P(Plant Tomorrow) = 1 − 0.74 = 0.26

(iii) This is a Bayes Theorem problem. We are given that it rains tomorrow and need to find the probability that it rained today (R₁):

P(R₁|R₂) = `(P(R)_1 × P(R_2|R1))/(P(R_2))`

Using values from part (i):

P(R₁|R₂) = `(0.4 xx 0.8)/0.74`

= `0.32/0.74`

= `32/74`

= `P(R1|R2)`

= `16/37`

= 0.432

(iv) P(Both days) = P(R′₁|) × P(R′₂|R′₁)

P(R′₂|R′₁) = 1 − P(R′2|R′1) 

= 1 − 0.7 = 0.3

P(Both days) = 0.6 × 0.3 = 0.18