Could someone please correct my work.
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) Draw a tree diagram
(sorry if it looks confusing)
Percent = 100%
80%
R / / 17% R
/----------
| . \ 83 R (not)
|
| . 68% R
\ . /
\ ----------
R (not rain) . \
20% . 32% R(not)
b) What is the probability that it rained today and a fish was caught?
P(RR) = 0.20 x 0.68 = 0.136
c) What is the probability of catching a fish today?
P(R not R|R) = 0.20 x 0.68 = 0.136
d) IIf a person caught a fish, what is the probability that there was rain?
I don't get this question*
Thanks
R is 20%, not rain 32% and R = 68%
To correct your work:
A) The tree diagram you drew is mostly correct. However, there is one small mistake in the branch probabilities for catching a fish at Lake Blue. In the branch where it doesn't rain (R (not)), the probability should be 83% (not 83 R). The corrected tree diagram looks like this:
```
100%
__________|__________
| |
80% 17%
(R) (not R)
| |
_______|_______ _______|_______
| | | |
68% 32% 68% 32%
(R) (not R) (R) (not R)
| | | |
20% 80% 17% 83%
(Caught) (Not Caught) (Caught) (Not Caught)
```
b) The probability that it rained today and a fish was caught is the product of the probabilities along the branch: P(RR) = 0.20 (probability of rain) x 0.68 (probability of catching a fish given rain) = 0.136 or 13.6%.
c) The probability of catching a fish today is the sum of the probabilities of catching a fish given rain and catching a fish given no rain:
P(Caught) = P(R) x P(Caught | R) + P(not R) x P(Caught | not R)
= 0.20 (probability of rain) x 0.68 (probability of catching a fish given rain) + 0.80 (probability of no rain) x 0.17 (probability of catching a fish given no rain)
= 0.136 + 0.136
= 0.272 or 27.2%.
d) The question asks for the probability that there was rain given that a person caught a fish. This can be calculated using Bayes' Theorem:
P(Rain | Caught) = P(Caught | Rain) x P(Rain) / P(Caught)
= 0.68 (probability of catching a fish given rain) x 0.20 (probability of rain) / 0.272 (probability of catching a fish)
= 0.125 or 12.5%.
So, if a person caught a fish, there is a 12.5% chance that there was rain.