There are 4 black marbles and 5 red marbles in a bag. If you reach in and randomly select 2 marbles, what is the probability that both are red? Explain your reasoning.

First marble = 5 out of 9 chances (5 reds, total of 9 marbles)

Second marble = 4 out of 8 chances (4 reds, total of 8 marbles left)

Probability = 5/9 x 1/2 = 5/18 = 27.78%

To find the probability that both marbles selected are red, we need to calculate the favorable outcomes (selecting two red marbles) and divide it by the total number of possible outcomes.

First, let's determine the total number of possible outcomes. When selecting two marbles from a bag, we consider the order as unimportant, which means we are dealing with combinations, not permutations. The formula to calculate combinations is:

nCr = n! / (r! * (n-r)!)

where n is the total number of items and r is the number of items we want to select.

In this case, we have a total of 9 marbles (4 black marbles + 5 red marbles) in the bag, and we want to select 2 marbles. Plugging these values into the formula, we get:

9C2 = 9! / (2! * (9-2)!)
= (9 * 8 * 7!) / (2 * 7!)
= (9 * 8) / 2
= 72 / 2
= 36

So, there are 36 possible outcomes when selecting two marbles from the bag.

Now, let's determine the number of favorable outcomes, i.e., the number of ways to select two red marbles. We have 5 red marbles in the bag, and we want to select 2. Using the same formula as before:

5C2 = 5! / (2! * (5-2)!)
= (5 * 4 * 3!) / (2 * 3!)
= (5 * 4) / 2
= 20 / 2
= 10

Therefore, there are 10 favorable outcomes where both marbles selected are red.

Finally, we can calculate the probability by dividing the favorable outcomes by the total possible outcomes:

Probability = Favorable outcomes / Total outcomes
= 10 / 36
= 5 / 18

So, the probability that both selected marbles are red is 5/18 or approximately 0.2778.