Question

An urn contains 5 blue marbles and 4 yellow marbles. One marble is​ removed, its color​ noted, and not replaced. A second marble is removed and its color is noted.

​(a) What is the probability that both marbles are blue? yellow​?

​(b) What is the probability that exactly one marble is blue​?

A tree diagram has a root that splits into 2 branches labeled blue and yellow. Each primary branch splits into 2 secondary branches, labeled blue and yellow.yellowblueblueblueyellowyellow

Answer 1

Since this is a case of sampling without replacement, so after a ball is sampled, the probabilities for other sampling is changed.

(a)

P(both marbles are blue) = P(first marble is blue)*P(second marble is blue)

P(first marble is blue) = Total blue marbles/Total marbles = 5/(5+4) = 5/9

After a blue marble is drawn, so there are only 8 left. So for the next time we have:

P(second marble is vlue) = Total blue marbles left/Total marbles left = 4/(4+4) = 1/2

So,

P(both marbles are blue) = (5/9)*(1/2) = 0.278

Similarly,

P(both marbles are yellow) = P(first marble is yellow)*P(second marble is yellow) = (4/9)*(3/8) = 0.167

(b)

P(exactly one marble is blue) = P(first marble is blue and second is yellow) + P(first marble is yellow and second is blue)

Now,

P(first marble is blue and second is yellow) = (5/9)*(4/8) = 0.278

P(first marble is yellow and second is blue) = (4/9)*(5/8) = 0.278

So,

P(exactly one marble is blue) = 0.278+0.278 = 0.556