Bayes formula and Movie critic

Question

I have the following exercice to solve in order to understand more conditional probabilities. Consider a movie critic who have the following ratios:

• 95% of the movies I liked are recommended by the critic
• 95% of the movies I disliked are not recommended by the critic
• I like in general 1% of the movies

Now knowing that the critic likes a movie, what is the probability I like it too?

In order to solve the exercice, I supposed that I have 2 variables A & B:

1. A can take 1 if I like the movie and if 0 dislike it.
2. B also takes 1 if the critic likes the movie and 0 if he dislikes it

Using Bayes's formula, I tried to calculate P(A=1\B=1)=P(B=1\A=1)*P(A)/P(B=1) which is the same as P(A=1\B=1)=P(A=1 & B=1)/P(B=1)

Now we know that P(A=1 & B=1)=0.95. The diffculty I am having now is in the calculation of P(B=1).

Any Hints?

Answer 1

HINT

Use:

$$P(B) = P(B \cap A) + P(B \cap A^C) = P(B|A) \cdot P(A) + P(B|A^C) \cdot P(A^C)$$

Also:

$$P(B|A^C) = 1-P(B^C|A^C)$$

(You're given $P(B|A)$ and $P(B^C|A^C)$, and I am sure you can figure out $P(A^C)$ yourself!)