A fishing magazine collected data and printed the following results:
If there is rain then the probability of catching a fish at Lake Blue is 80%, but if it doesn't rain then the probability of catching a fish at Lake Blue is 17%. The forecast for today is 68% chance of rain.
a) What is the probability that it rained today and a fish was caught?
b) What is the probability of catching a fish today?
P(R not R|R) = 0.20 x 0.68 = 0.136
c) If a person caught a fish, what is the probability that there was rain?
To calculate the probability that it rained today and a fish was caught, we need to multiply the probability of rain (0.68) by the probability of catching a fish given that it rains (0.80):
a) P(Rain and Fish) = P(Rain) × P(Fish|Rain) = 0.68 × 0.80 = 0.544
So, the probability that it rained today and a fish was caught is 0.544, or 54.4%.
To calculate the probability of catching a fish today, we need to consider both the probability of catching a fish given that it rains and the probability of catching a fish given that it doesn't rain.
b) P(Fish) = P(Rain) × P(Fish|Rain) + P(Not Rain) × P(Fish|Not Rain)
= 0.68 × 0.80 + 0.32 × 0.17
= 0.544 + 0.0544
= 0.5984
So, the probability of catching a fish today is 0.5984, or 59.84%.
To calculate the probability that there was rain given a person caught a fish, we can use Bayes' Theorem. We can use the formula:
P(Rain|Fish) = (P(Fish|Rain) × P(Rain)) ÷ P(Fish)
c) P(Rain|Fish) = (P(Fish|Rain) × P(Rain)) ÷ P(Fish)
= (0.80 × 0.68) ÷ 0.5984
= 0.544 ÷ 0.5984
≈ 0.9091
So, the probability that there was rain given a person caught a fish is approximately 0.9091, or 90.91%.
To solve these probabilities, we will use conditional probability and multiply the probability of two events happening together.
a) To find the probability that it rained today and a fish was caught, we can use the formula:
P(Rain and Fish caught) = P(Fish caught | Rain) x P(Rain)
We are given:
P(Fish caught | Rain) = 0.80 (probability of catching a fish when it rains)
P(Rain) = 0.68 (probability that it rained today)
Therefore, the probability that it rained today and a fish was caught is:
P(Rain and Fish caught) = 0.80 x 0.68 = 0.544
b) To find the probability of catching a fish today, we can use the formula for total probability:
P(Fish caught) = P(Fish caught | Rain) x P(Rain) + P(Fish caught | No Rain) x P(No Rain)
We are given:
P(Fish caught | Rain) = 0.80 (probability of catching a fish when it rains)
P(Rain) = 0.68 (probability that it rained today)
P(Fish caught | No Rain) = 0.17 (probability of catching a fish when it doesn't rain)
P(No Rain) = 1 - P(Rain) = 1 - 0.68 = 0.32 (probability that it didn't rain today)
Therefore, the probability of catching a fish today is:
P(Fish caught) = 0.80 x 0.68 + 0.17 x 0.32 = 0.544 + 0.0544 = 0.5984
c) To find the probability that there was rain given that a fish was caught, we can use Bayes' theorem:
P(Rain | Fish caught) = (P(Fish caught | Rain) x P(Rain)) / P(Fish caught)
We know:
P(Fish caught | Rain) = 0.80 (probability of catching a fish when it rains)
P(Rain) = 0.68 (probability that it rained today)
P(Fish caught) = 0.5984 (probability of catching a fish today, as calculated in part b)
Therefore, the probability that there was rain given that a fish was caught is:
P(Rain | Fish caught) = (0.80 x 0.68) / 0.5984 = 0.544 / 0.5984 ≈ 0.908