We have two urns, where urn-1 contains 3 blue balls and 1 red ball while urn-2 contains 2 blue balls and 2 red balls. Suppose we will randomly select one of the two urns, and then randomly pick a ball from the selected urn. a) What is the probability that we will pick a blue ball?
b) If we picked a blue ball, what is the probability that the selected urn was urn-1?
c) Suppose we picked a blue ball. If we randomly pick one additional ball from the same urn, what is the probability that we pick a red ball?

a) What is the probability that we will pick a blue ball?

there is a 50% chance that urn 1 will be selected and the possibility of a blue ball is 3/4 x 0.5 = 1.5/4

there is also a 50% chance that urn 2 will be selected and the possibility of a blue ball is 2/4 x 0.5 = 1/4

the probability of choosing a blue ball = 1.5/4 + 1/4 = 2.5/4 or 5/8

b) If we picked a blue ball, what is the probability that the selected urn was urn-1?

there are 5 blue balls in total, and 3 of them come from urn 1

c) Suppose we picked a blue ball. If we randomly pick one additional ball from the same urn, what is the probability that we pick a red ball?

there is a 50% chance that urn 1 will be selected, and the possibility of a red ball after a blue ball is 1/3 x 0.5 = 0.5/3

there is also a 50% chance that urn 2 will be selected and the possibility of a red ball after a blue ball is 2/3 x 0.5 = 1/3

the possibility of choosing a red ball after a blue ball = 0.5/3 + 1/3 = 1.5/3 = 1/2

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3