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.

This is assuming no replacement.

First red marble = 5/9

Second red marble = 4/8

With replacement, the second red marble = 5/9

The probability of both/all events occurring is found by multiplying the individual events.

To find the probability of drawing two red marbles from the bag, we need to determine the total number of possible outcomes and the number of favorable outcomes.

Total number of possible outcomes:
When we randomly select 2 marbles from the bag, there are a total of 9 marbles to choose from.

Number of favorable outcomes:
Out of the 9 marbles, we want to select 2 red marbles. Since there are 5 red marbles in the bag, we can select 2 out of 5 in C(5, 2) ways.

The combination formula used above is given by C(n, r) = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items we want to select.

So, C(5, 2) = 5! / (2! * (5-2)!) = 5! / (2! * 3!) = (5 * 4 * 3!) / (2! * 3!) = 10.

Therefore, there are 10 favorable outcomes.

Now, we can find the probability by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= 10 / 9
= 10/9 or approximately 1.11.

So, the probability that both marbles selected are red is approximately 1.11 or 10/9.

To find the probability of randomly selecting 2 red marbles from the bag, we need to find the number of favorable outcomes (selecting two red marbles) and divide it by the total number of possible outcomes (selecting any two marbles).

First, let's determine the total number of marbles in the bag. We have 4 black marbles and 5 red marbles, so the total number of marbles is 4 + 5 = 9.

To find the number of favorable outcomes (selecting two red marbles), we need to count the number of ways we can choose 2 out of the 5 red marbles. We can use the combination formula, which is given by:

C(n, r) = n! / (r! * (n-r)!)

Where "n" represents the total number of items to choose from, and "r" represents the number of items we want to choose.

In our case, n = 5 (total number of red marbles) and r = 2 (number of red marbles we want to choose).

Using the combination formula, we can calculate:

C(5, 2) = 5! / (2! * (5-2)!)
= (5 * 4 * 3 * 2 * 1) / ((2 * 1) * (3 * 2 * 1))
= (120) / (2 * 6)
= 10

So, there are 10 different ways we can choose 2 red marbles from the bag.

Next, let's determine the total number of possible outcomes (selecting any two marbles). We can use the combination formula again, but this time n = 9 (total number of marbles) and r = 2 (number of marbles we want to choose).

Using the combination formula:

C(9, 2) = 9! / (2! * (9-2)!)
= (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * (7 * 6 * 5 * 4 * 3 * 2 * 1))
= (9 * 8) / (2 * 1)
= 36

So, there are 36 different ways we can choose any 2 marbles from the bag.

To find the probability, we divide the number of favorable outcomes (10) by the total number of possible outcomes (36):

Probability of selecting 2 red marbles = 10 / 36
= 5 / 18

Therefore, the probability of randomly selecting 2 red marbles from the bag is 5/18.