a. probability that second marble is violet b. probability that both marbles are the same colors c. draw a tree diagram

Question

An urn contains 5 red and 8 violet marbles. One marble is drawn from the urn and it will be replaced by two marbles of the other colour. Then a second marble is drawn.

Answer 1

Let's think about the events taking place here:

 

All the events that the second marble is violet can be subdivided into two categories: the first marble was violet and the second was violet, and the first marble was red and the second was violet. Since these two events are distinct and complimentary, we can add their probabilities to get the probability of just the 2nd marble being violet.

 

P(2nd = violet) = P(2nd = violet and 1st = red)  +  P(2nd = violet and 1st = violet)

 

Now we can apply the definition of conditional probability to each. [Read the "|" as "given"]

 

P(2nd = violet AND 1st = red) = P(2nd = violet | 1st = red) * P(1st = red)

 

P(2nd = violet and 1st = violet) = P(2nd = violet | 1st = violet) * P(1st = violet)

 

Note that P(1st = red) = 5/13 and P(1st = violet) = 8/13. 

 

From the statement above, we know that if the first marble was red, then two violet marbles replace it, so there are 4 reds and 10 violets for the second draw. Hence P(2nd = violet | 1st = red) = 10/14.

 

From the statement above, we know that if the first marble was violet, then two red marbles replace it, so there are 7 reds and 7 violets for the second draw. Hence P(2nd = violet | 1st = violet) = 1/2.

 

Combining all of these probabilities, we get: 

 

P(2nd = violet) = 10/14 * 5/13   +   1/2*8/13 = 0.58