Question

You reach into a bag that contains the following distribution of colored marbles: 5 green, 8 yellow, 3 purple. You will grab 2 marbles (one at a time) and leave them on the table in front of you. a. (1.5pts) Find the probability that you first select a green and second a yellow marble. b. (2pts) Find the probability that you select no green marbles in your two picks. c. (2pts) Find the probability that you select exactly 1 purple marble and one non-purple marble.

Answer 1

a. The probability that you first select a green and second a yellow marble is 1/6

b. The probability that you select no green marbles in your two picks is 11/24

c. The probability that you select exactly 1 purple marble and one non-purple marble is 13/40

Explanation:

We have 5 green marbles, 8 yellow marbles, and 3 purple marbles,

Which gives a total of,

16 marbles, Now we calculate the probabilities,

a. (1.5pts) Find the probability that you first select a green and second a yellow marble.

The probability of selecting a green marble is, 5/16

Then, 15 marbles will be left out of which 8 are yellow,

so the probability of selecting a yellow marble from the 15 remaining marbles will be,8/15

The total probability will be the product of these 2, so we get,

P = (5/16)(8/15) = P = 1/6

So, the probability that you first select a green and second a yellow marble is 1/6

b. (2pts) Find the probability that you select no green marbles in your two picks

Since there are 5 green marbles, the probability that we don't pick the green marble out of the total 16 marbles is, on the first try,

The probability of picking the green one is 5/16 so, for not picking we get,

1 - 5/16 = 11/16

Now, for the 2nd try, there are still 5 green marbles but the total number of marbles is now 15, the probability of picking a green is then 5/15 = 1/3

and of not picking a green is then, 1 - 1/3 = 2/3

The total probability is then the product

P = (11/16)(2/3) = 11/24

So, the probability that you select no green marbles in your two picks is 11/24

c. (2pts) Find the probability that you select exactly 1 purple marble and one non-purple marble

Here there are two cases, either we get the purple on our first try with 16 marbles or the 2nd try with 15 marbles,

in the first case,

The probability of getting a purple is 3/16

And then there are 2 purples left and 15 total marbles

So, the 2nd one is not purple, so the probability of not getting the purple on the 2nd try is 1 - 2/15 = 13/15

And we get the product,

(3/16)(13/15) = 13/80

It can also happen that the first one is not purple and the 2nd one is purple in that case,

The probability that 1st one is not purple = 1 - 3/16 = 13/16

The 2nd one is purple = 3/15

And the product is,

(13/16)(3/15) = 13/80

Combining these 2 cases to get the total probabilty of getting exactly 1 purple marble we have,

13/80 + 13/80 = 13/40 is the probability of getting exactly one purple.