Question

betty picks 4 random marbles from a bowl containing 3 white, 4 yellow, and 5 blue marbles. the probability that exactly 1 of the 4 marbles drawn is blue is . the probability that at least 1 of the 4 marbles drawn is white is .

Answer 1

There are 12 marbles in total in the bowl.
From this bowl, we will choose 4 marbles at random.
That's 12C4 or (12*11*10*9) or 11,880 ways

Out of this, what is the probability that there is exactly 1 blue marble.
There are 5 blue balls but only 1 is selected, 5C1 ways
The rest except blue calls are 7, but only 3 slots are left, 7C3 ways

The probably of exactly 1 blue of 4 marbles is then 5C1*7C3/12C4 = 1050/11,880 = 0.0884 or 8.84%

Answer 2

I know this is two years later, but I figure it is worth helping others out with this challenging question. Here's what I got: 0.353 and 0.7455.

The first part, I got 0.3535. There are 5 blue marbles and 7 not-blue marbles. We need the probability of getting a selection of 4 marbles with one blue marble. This means we're going to have to use nCr.

Betty's picking 1 blue marble out of 5 blue marbles for this theoretical selection, so we have 5c1. For the other marbles, she's picking 3 out of 7 not-blue marbles, so we have 7c3. Finding the total combinations for picking 1 blue marble out of 5 blue marbles AND picking 3 out of 7 not-blue marbles is 5c1 x 7c3. Put that over the total, 12c4 (4 marbles out of the total selections), and you get 0.3535. I hope this is correct :)

For the second part, we have the same concept but with three steps. We have 3 whites and 9 non-whites. We're going to add the probability of Betty picking out 1 white and 3 nons, 2 whites and 2 nons, OR 3 whites and 1 nons.

For the first probability, we have: (3c1 x 9c3) / 12c4

For the second probability, we have: (3c2 x 9c2) / 12c4

For the third and final probability, we have: (3c3 x 9c1) / 12c4

Add these altogether and you get 0.74545 (0.7455). For all of these calculations, I've used a calculator; doing them all by hand is quite daunting. Hope this helps for anyone who needs it!