Bayes Theorem

P(B/A)=P(A and B)/P(A) but from Bayes Theorem we have

P(B/A) = P(A/B)*P(B)/P(A)

P(B/A) = P(A/B)*P(B)/P(A)

in our problem let A = identify correctly and B = cat person

therefore A' = identify incorrectly and B' = dog person

therefore A' = identify incorrectly and B' = dog person

P(A) = P(A/B)*P(B) + P(A/B')P(B')

Note the tree diagram in the written work.

The values used and obtained are as follows

P(B) = .33

P(B') = .67

P(A/B) = .96

P(A'/B) = .04

P(A/B') = .71

P(A'/B') = .29

P(A and B) = .33(.76) = .3168

P(A' and B) = .33(.04) = .0132

P(A and B') = .67(.71) = .4757

P(A' and B') = .67(.29) = .1943

P(B') = .67

P(A/B) = .96

P(A'/B) = .04

P(A/B') = .71

P(A'/B') = .29

P(A and B) = .33(.76) = .3168

P(A' and B) = .33(.04) = .0132

P(A and B') = .67(.71) = .4757

P(A' and B') = .67(.29) = .1943

Notice that all the joint probabilities add to 1

Now put those values into the formula and you'll get P(B/A) = .3997