Question

According to the law of total probability the event space is partitioned into a disjoint set of hypotheses. As a result, the evidence is also split across these hypotheses. Going back to our example, the space of people who Like Burger King is actually part of a bigger space consisting of people who Like Burger King and Do Not Like Burger King. In fact, if you look at our table you see both columns. Given this idea, the denominator for our Bayes' theorem P(B) can actually be split into P(B∩A)+P(B∩notA). Both probabilities are added because they constitute the entire space of P(B). P(A|B)=P(B|A)P(A)P(B)=P(B|A)P(A)P(B∩A)+P(B∩notA) Furthermore, we can apply the conditional probability formula to transform the intersections P(B∩A) and P(B∩notA) so that. P(A|B)=P(B|A)P(A)P(B)=P(B|A)P(A)P(B|A)P(A)+P(B|notA)P(notA) Like Burger King Not Like Burger King Total Like Whopper 5 2 7 Not Like Whopper 1 2 3 Total 6 4 10 Using our table we apply Bayes' theorem with the law of total probability P(LikeBurgerKing|LikeWhopper)=P(LikeWhopper|LikeBurgerKing)P(LikeBurgerKing)P(LikeWhopper|LikeBurgerKing)P(LikeBurgerKing)+P(LikeWhopper|notLikeBurgerKing)P(notLikeBurgerKing=56⋅61056⋅610+24⋅410=1212+15=12710=12⋅107=57=0.71 As expected, the result is again the same as the value we got from conditional probability and the Bayes' theorem because they are all just variations. Your turn! Using the table below, apply Bayes' theorem with the law of total probability to get the probability of people who like Burger King given that they like french fries. Like Burger King Like French Fries Y Y N Y Y N Y Y Y N N Y N N N Y Y Y Y Y Answer:

Answer 1

Y Y N Y Y N Y Y Y N N Y N N N Y Y Y Y Y

No. of yeses for both = 13

No. of nos for burger but yes for fries= 7

Total number of subjects = 20 (13 + 7)

= ( likes fries irrespective of liking burger)

= 13 / 20

Ans: 0.65