conditional probability question

[emlmnop]

hey, i've been struggling to find p(B) in this question

the chance of a bushfire is 85% after a period of no rain and 21% after a period of rain. the chance of rain is 46%. find the probability that:

let p(a) be the probability of it raining and p(b) be the probability of a bushfire

a) there is not a bushfire given it has rained

p(B'|A) = 1-P(B|A)
             = 0.79

b) it has rained given there is a bushfire

P(A|B) = P(A and B) / P(B) (how do you find p(B)?)

c) it has not rained given there is a bushfire

P(A'|B) = 1 - P(A|B)

d) it has rained given there is not a bushfire

P(A|B')

[1729]

hey, i've been struggling to find p(B) in this question

the chance of a bushfire is 85% after a period of no rain and 21% after a period of rain. the chance of rain is 46%. find the probability that:

let p(a) be the probability of it raining and p(b) be the probability of a bushfire

a) there is not a bushfire given it has rained

p(B'|A) = 1-P(B|A)
             = 0.79

b) it has rained given there is a bushfire

P(A|B) = P(A and B) / P(B) (how do you find p(B)?)

c) it has not rained given there is a bushfire

P(A'|B) = 1 - P(A|B)

d) it has rained given there is not a bushfire

P(A|B')

Hi there! Welcome to the forums!
Part b is just bayes rule + law of total probability, its easiest to visualize/understand if you draw a tree of the different probabilities and outcomes.
This is from the given information:

So the probability that there is a bushfire is just the sum of the two branches that end in "bushfire", or P(rain)P(bushfire | rain) + P(no rain)P(bushfire | no rain), which is the law of total probability. You can think of it like this: a bushfire can happen under only two cases; either there is rain, or there is no rain, the probability of a bushfire happening is the sum of the two probabilities: P(bushfire and rain) + P(bushfire and no rain) and this is equal to P(bushfire | rain)P(rain) + P(bushfire | no rain)P(no rain), by rearranging the definition of conditional probability.

If you want a formula (which i generally don't like because rote memorization isn't the best way to learn things)


Hope this helps!

Don't hesitate to ask any questions.

EDIT: Oops - In that picture, that 0.34 should be a 0.54 on the left

[emlmnop]

Hi there! Welcome to the forums!
Part b is just bayes rule + law of total probability, its easiest to visualize/understand if you draw a tree of the different probabilities and outcomes.
This is from the given information:
(Image removed from quote.)
So the probability that there is a bushfire is just the sum of the two branches that end in "bushfire", or P(rain)P(bushfire | rain) + P(no rain)P(bushfire | no rain), which is the law of total probability. You can think of it like this: a bushfire can happen under only two cases; either there is rain, or there is no rain, the probability of a bushfire happening is the sum of the two probabilities: P(bushfire and rain) + P(bushfire and no rain) and this is equal to P(bushfire | rain)P(rain) + P(bushfire | no rain)P(no rain), by rearranging the definition of conditional probability.

If you want a formula (which i generally don't like because rote memorization isn't the best way to learn things)


Hope this helps!

Don't hesitate to ask any questions.

EDIT: Oops - In that picture, that 0.34 should be a 0.54 on the left

Thank you! Can you help me out w part d as well please?

[keltingmeith]

Thank you! Can you help me out w part d as well please?

Part d is just another application of Bayes' rule - why don't you try doing as 1729 has already shown you, but this time on P(A|B')