Help with finte math problem

Question

An urn contains 4 red balls and 3 green balls. Two balls are drawn randomly at the same time. Let m = the event that the balls are the same color and let R = tje event that both balls are drawn are red. Find the conditional probability Pr(R|M) and Pr(M|R). Simplify as far as possible.

Answer 1

4 red, 3 green. randomly choose two

if both are the same color what is the probability they are both red?

if both are red, what is the probability both are the same color? that's 100% obviously, with no need for calculations. if they're both red, they're both the same color

but if both are the same color, they could both be red or both green

this requires some calculation

although it's (obvious) there's a greater chance both are red, so probability is greater than 50%

but not that much greater as 4 compared to 3 is just one ball more of red than green

a tree diagram might help, call the balls R1,R2,R3,R4 and G1,G2,G3,

branches are

R1R2

R1R3

R1R4

R2R3

R2R4

R3R4

G1G2

G1G3

G2G3

9 branches, 6 both red, 3 both green

twice as many are red

so 66 2/3 % chance both red

33 1/3 % both green

P(r/m)= 66.666..%

P(m/r) = 100%