A hat contains 3 ​marbles; one green​, one blue​, and one orange.

Two marbles are selected at random without replacement from the hat.

​a) Determine the number of points in the sample space.

​b) Determine the probability of selecting exactly 1 green marble.

​c) Determine the probability of selecting at least 1 marble that is not green.

​d) Determine the probability of selecting no orange marbles.

To answer these questions, we need to understand the concept of sample space and probability.

a) The sample space is the set of all possible outcomes of an experiment. In this case, we are selecting 2 marbles from a hat containing 3 marbles. The order of selection does not matter, and we are selecting without replacement. Therefore, the number of points in the sample space can be determined using the formula for combinations.

Since there are 3 marbles in the hat, we can select any 2 of them. The number of ways to select 2 marbles out of 3 is determined by the combination formula, which is denoted as nCr, where n is the number of objects and r is the number to be chosen.

Using the formula, we have:
3C2 = 3! / (2!(3-2)!) = 3.

Therefore, the number of points in the sample space is 3.

b) To determine the probability of selecting exactly 1 green marble, we need to count the favorable outcomes and divide it by the total number of outcomes.

Favorable outcomes: There are two ways to select exactly 1 green marble: selecting the green marble first and the non-green marble second, or selecting the non-green marble first and the green marble second. Therefore, we have 2 favorable outcomes.

Total outcomes: We already determined that the sample space has 3 points, so we have 3 total outcomes.

Probability = Favorable outcomes / Total outcomes = 2 / 3.

Therefore, the probability of selecting exactly 1 green marble is 2/3.

c) To determine the probability of selecting at least 1 marble that is not green, we need to count the favorable outcomes and divide it by the total number of outcomes.

Favorable outcomes: In this case, favorable outcomes include selecting the blue marble, selecting the orange marble, or selecting both the blue and orange marbles. There are 3 favorable outcomes.

Total outcomes: We already determined that the sample space has 3 points, so we have 3 total outcomes.

Probability = Favorable outcomes / Total outcomes = 3 / 3 = 1.

Therefore, the probability of selecting at least 1 marble that is not green is 1.

d) To determine the probability of selecting no orange marbles, we need to count the favorable outcomes and divide it by the total number of outcomes.

Favorable outcomes: Since we are selecting without replacement, there is only one favorable outcome: selecting the blue marble first and the non-orange marble second. Therefore, we have 1 favorable outcome.

Total outcomes: We already determined that the sample space has 3 points, so we have 3 total outcomes.

Probability = Favorable outcomes / Total outcomes = 1 / 3.

Therefore, the probability of selecting no orange marbles is 1/3.