I really need help in the below question.

Thanks

Kingman

In a test, questions are picked randomly. The probability that a participant gets an easy question is .6 while the probability of getting a difficult question is .4.

The probability of any participant giving the correct answer to an easy question is .8 while the probability

of giving the correct answer to a difficult question is .3.

Find the probability that the participant give the correct answer twice in a row .

I have 2 approaches to the question but wonder how to prove that the solution are the same using Venn diagram. Let

E: an event that participant picked an easy question.

D: an event that participant picked a difficult question.

C: an event that the participant give the correct answer,

W: an event that the participant give the wrong answer.

Solution 1:

Probability = (.6*.8+.4*.3)2 =.36 (using product rule)

Solution 2:

Using Tree diagram,

Probability = .6*.8 * (.6*.8+.4*.3) + .4*.3 * (.6*.8+.4*.3) =.36