Question

A box contains 4 red and 7 green marbles. You reach in and remove 3 marbles all at once.

(a) Find the probability that these 3 marbles are all red. (Enter your answer as a fraction.)

(b) Find the probability that these 3 marbles are all of the same color. (Enter your answer as a fraction.)

*Please show your work so I know how you got your answers. TY

Answer 1

a)

3 red marbles are selected from 4 red marbles by ⁴C₃ ways.

Where ⁿC_r = n! / [ (n-r)! * r! ]

Therefore, ⁴C₃ = 4! / [(4-3)! * 3! ] = 4

Total number of ways to select 3 marbles from (4+7 = 11) 11 marbles =  ¹¹C₃

Therefore,  

P( All three are red ) = ⁴C₃ / ¹¹C₃

= 4 / 165

b)

All 3 marbles are of same color = All three marbles are red OR all three marbles are green.

P(All 3 marbles are of same color) = P(All three marbles are red) + P(all three marbles are green)

We calculated P(All three marbles are red) = 4 / 165 in part a)

Now

3 marbles are selected from 7 green marbles by ⁷C₃ ways.

Also . total number of ways to select 3 marbles from (4+7) =11 marbles = ¹¹C₃

So, P(All three marbles are red) = ⁷C₃ / ¹¹C₃

= 35 / 165

Therefore,

P(All 3 marbles are of same color) = 4 / 165 + 35 / 165

= 39 / 165

= 13 / 55