ANNIKA HAS A SOCK FILLED WITH 3 RED MARBLES AND 5 GREEN MARBLES. WHAT IS THE PROBABILITY OF HER CHOOSING TWO RED MARBLES WITH AND WITHOUT REPLACEMENT?

with replacement

Pr(R,R)=3/8*3/8
without replacement
Pr(R,R)=3/8*2/7

To find the probability of Annika choosing two red marbles with and without replacement, we need to consider the total number of marbles and the number of red marbles.

Step 1: Calculate the probability of choosing two red marbles with replacement.

With replacement means that after each selection, the marble is replaced back into the sock.

Total number of marbles = 3 (red) + 5 (green) = 8
Probability of choosing a red marble on the first draw = Number of red marbles / Total number of marbles = 3/8

Since the marbles are replaced after each draw, the probability of choosing another red marble on the second draw is also 3/8.

The probability of choosing two red marbles with replacement is calculated by multiplying the probabilities of each draw:

Probability of choosing two red marbles with replacement = (3/8) * (3/8) = 9/64

Step 2: Calculate the probability of choosing two red marbles without replacement.

Without replacement means that after each selection, the marble is not replaced back into the sock.

Total number of marbles = 3 (red) + 5 (green) = 8
Probability of choosing a red marble on the first draw = Number of red marbles / Total number of marbles = 3/8

After one red marble is chosen, there are now 2 red marbles left and a total of 7 marbles remaining.
Probability of choosing a red marble on the second draw = Number of red marbles left / Total number of marbles left = 2/7

The probability of choosing two red marbles without replacement is calculated by multiplying the probabilities of each draw:

Probability of choosing two red marbles without replacement = (3/8) * (2/7) = 6/56 = 3/28

Therefore, the probability of Annika choosing two red marbles with replacement is 9/64, and without replacement is 3/28.

To find the probability of Annika choosing two red marbles with replacement, we first need to determine the total number of marbles in the sock. In this case, she has 3 red marbles and 5 green marbles, so the total number of marbles is 3 + 5 = 8.

With replacement means that after each draw, the marble is returned to the sock, so the total number of marbles remains the same. Since there are 3 red marbles out of 8 total marbles, the probability of choosing a red marble on the first draw is 3/8.

Since the marble is returned to the sock after each draw, the probability of choosing a red marble on the second draw is also 3/8, regardless of the outcome of the first draw.

To find the probability of both events happening (choosing two red marbles with replacement), we multiply the probabilities of each event. So the probability of choosing two red marbles with replacement is (3/8) * (3/8) = 9/64.

To find the probability of Annika choosing two red marbles without replacement, we need to account for the fact that after each draw, the marble is not returned to the sock.

The probability of choosing a red marble on the first draw is still 3/8, as there are 3 red marbles out of 8 total marbles. However, after the first draw, there are now 2 red marbles left out of 7 total marbles.

Therefore, the probability of choosing a red marble on the second draw (without replacement) is 2/7.

To find the probability of both events happening (choosing two red marbles without replacement), we multiply the probabilities of each event. So the probability of choosing two red marbles without replacement is (3/8) * (2/7) = 6/56, which simplifies to 3/28.