Respuesta :
First an intro to conditional probability.
This is the definition of conditional probability.
If x and y are statements then [tex]P(x|y)=\frac{P(x\cap y)}{P(y)}[/tex].
That is, the probability of x given y is the probability of x and y over the probability of y.
If we let the event that there is perfect conditions be y and the event that you catch a fish be x then,
P(y) = 0.76
P(x|y)=0.84
P(~x|~y)=0.07 (tildes represent "not")
From here by the definition of conditional probability P(x) = P(x|y)P(y)+P(x|~y)P(~y)
Notice also that the two term being added are the probability that the conditions are perfect and you catch something and the probability that the conditions are not perfect and you catch something. Since the conditions can either be perfect or not perfect the expression gives the correct result.
Now P(x|y)P(y)+P(x|~y)P(~y)=P(x|y)P(y)+(1-P(~x|~y))(1-P(y)).
Remember, that if c is an event P(~c) = 1-P(c). That is the probability that something happens is one minus the probability that it doesn't happen.
Plugging stuff in we get 0.8616.
For the next problem, we use Bayes' rule. It says that if a and b are events
[tex]P(a|b)=\frac{P(b|a)P(a)}{P(b)}[/tex].
Then, [tex]P(y|x)=\frac{P(x|y)P(y)}{P(x)}[/tex].
Using our answer from a, we get 0.7409.
