Question

A person has undertaken a mining job. The probabilities of completion of job on time with and without rain are 0.42 and 0.90 respectively. If the probability that it will rain is 0.45, then determine the probability that the mining job will be completed on time.
1. 0.684
2. 0.854
3. 0.554
4. 0.764

Answer 1

Correct Answer - Option 1 : 0.684

Calculation:

⇒ Let A be the event that the mining job will be completed on time and B be the event that it rains. We have,

⇒ P(B) = 0.45,

⇒ P(no rain) = P(B′) = 1 − P(B) = 1 − 0.45 = 0.55

⇒ By multiplication law of probability,

⇒ P(A|B) = 0.42

⇒ P(A|B′) = 0.90

Since, events B and B′ form partitions of the sample space S, by total probability theorem, we have

⇒ P(A) = P(B) P(A|B) + P(B′) P(A|B′)

⇒ 0.45 × 0.42 + 0.55 × 0.9

⇒ 0.189 + 0.495 = 0.684

⇒ So, the probability that the job will be completed on time is 0.684.

⇒ Total Probability Theorem

⇒ Given  mutually exclusive events A₁, A₂ …….. A_n whose probabilities sum to unity, then

⇒ P(B) = P(B|A₁)P(A₁) + P(B|A₂)P(A₂) + ………. + P(B|A_n)P(A_n)

⇒ Where B is an arbitrary event, and P (B|A_i) is the conditional probability of B assuming A_i .