Question

There are ten balls in an urn. They are identical except for color. Five are red, four are blue, and one is yellow. You are to draw a ball from the urn, note its color, and set it aside. Then you are to draw another ball from the urn and note its color.

(b) Let P(x, y) be the probability of choosing an x-colored ball on the first draw and a y-colored ball on the second draw. Compute the probability for each outcome of the experiment. (Enter your answers as fractions.)

P(R, R) =
P(R, B) =
P(R, Y) =
P(B, R) =
P(B, B) =
P(B, Y) =
P(Y, R) =
P(Y, B) =

Answer 1

P(R,R) = P(first ball is red and second ball is red)

= P(first ball is red)*P(second ball is red | first ball is red)

= (5/10)*(4/9) [On the first draw, there are 10 balls in the urn out of which 5 are red. On the second draw, there are 9 balls in the urn out of which 4 are red]

= 2/9

P(R,B) = P(first ball is red and second ball is blue)

= P(first ball is red)*P(second ball is blue | first ball is red)

= (5/10)*(4/9) [On the first draw, there are 10 balls in the urn out of which 5 are red. On the second draw, there are 9 balls in the urn out of which 4 are blue]

= 2/9

P(R,Y) = P(first ball is red and second ball is yellow)

= P(first ball is red)*P(second ball is yellow | first ball is red)

= (5/10)*(1/9) [On the first draw, there are 10 balls in the urn out of which 5 are red. On the second draw, there are 9 balls in the urn out of which 1 is yellow]

= 1/18

P(B,R) = P(first ball is blue and second ball is red)

= P(first ball is blue)*P(second ball is red | first ball is blue)

= (4/10)*(5/9) [On the first draw, there are 10 balls in the urn out of which 4 are blue. On the second draw, there are 9 balls in the urn out of which 5 are red]

= 2/9

P(B,B) = P(first ball is blue and second ball is blue)

= P(first ball is blue)*P(second ball is blue | first ball is blue)

= (4/10)*(3/9) [On the first draw, there are 10 balls in the urn out of which 4 are blue. On the second draw, there are 9 balls in the urn out of which 3 are blue]

= 2/15

P(B,Y) = P(first ball is blue and second ball is yellow)

= P(first ball is blue)*P(second ball is yellow | first ball is blue)

= (4/10)*(1/9) [On the first draw, there are 10 balls in the urn out of which 4 are blue. On the second draw, there are 9 balls in the urn out of which 1 is yellow]

= 2/45

P(Y,R) = P(first ball is yellow and second ball is red)

= P(first ball is yellow)*P(second ball is red | first ball is yellow)

= (1/10)*(5/9) [On the first draw, there are 10 balls in the urn out of which 1 is yellow. On the second draw, there are 9 balls in the urn out of which 5 are red]

= 1/18

P(Y,B) = P(first ball is yellow and second ball is blue)

= P(first ball is yellow)*P(second ball is blue | first ball is yellow)

= (1/10)*(4/9) [On the first draw, there are 10 balls in the urn out of which 1 is yellow. On the second draw, there are 9 balls in the urn out of which 4 are blue]

= 2/45

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