Mathematics

Mathematics

Problem

During a 90-day weather study at a hill station, the following events were recorded for any randomly chosen day: R : "It rained" T : "Thunder was heard". It was found that P(R)=0.55P(R)=0.55, P(T)=0.30P(T)=0.30 and P(R∩T)=0.18P(R\cap T)=0.18. a. (i) Calculate the probability that it either rained or thunder was heard on a given day. b. (ii) Find the probability that it rained without any thunder on a given day. c. (iii) Determine the probability that a day experienced neither rain nor thunder.
Intuitive explanation

This problem is about understanding how events overlap using probability rules. The key insight is that P(R∩T)=0.18P(R \cap T) = 0.18 tells you the probability both rain and thunder occur together—which is smaller than either individual probability, showing they're not independent. You'll use this to find probabilities of combined events like "rain or thunder" using the addition rule, or conditional probabilities like "thunder given it rained" using the definition P(T∣R)=P(R∩T)P(R)P(T|R) = \frac{P(R \cap T)}{P(R)}.

Intuitive explanation

This problem is about understanding how events overlap using probability rules. The key insight is that P(R∩T)=0.18P(R \cap T) = 0.18 tells you the probability both rain and thunder occur together—which is smaller than either individual probability, showing they're not independent. You'll use this to find probabilities of combined events like "rain or thunder" using the addition rule, or conditional probabilities like "thunder given it rained" using the definition P(T∣R)=P(R∩T)P(R)P(T|R) = \frac{P(R \cap T)}{P(R)}.

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