a box contains the following mixture of colored marbles 2 black, 3 red, 5 yellow, and 2 green. If two marbles are drawn, the second being drawn n after the first is replaced, then what is the probability that both are black?

To find the probability of drawing two black marbles, we need to determine the total number of marbles and the number of black marbles.

Total number of marbles = 2 (black) + 3 (red) + 5 (yellow) + 2 (green) = 12 marbles

Number of black marbles = 2

When drawing with replacement, the probability of an event occurring remains the same each time. Therefore, the probability of drawing a black marble on the first draw is 2/12.

Since the marble is replaced, the probability of drawing a black marble on the second draw is also 2/12.

To find the probability of both events happening, we multiply the probabilities: (2/12) * (2/12) = 4/144 = 1/36.

Therefore, the probability of drawing two black marbles with replacement is 1/36.

To find the probability of drawing two black marbles, we need to calculate the probability of drawing a black marble on the first draw and then drawing a black marble again on the second draw, assuming the first marble is replaced.

Step 1: Find the probability of drawing a black marble on the first draw.
In this case, there are a total of 2 black marbles out of the total number of marbles in the box. Therefore, the probability of drawing a black marble on the first draw is:
P(1st draw = black) = number of black marbles / total number of marbles
P(1st draw = black) = 2 / (2 + 3 + 5 + 2)
P(1st draw = black) = 2 / 12
P(1st draw = black) = 1/6

Step 2: Find the probability of drawing a black marble on the second draw after replacing the first black marble.
Since we are replacing the first marble after it is drawn, the probability of drawing a black marble on the second draw is the same as the probability of drawing a black marble on the first draw:
P(2nd draw = black) = P(1st draw = black)
P(2nd draw = black) = 1/6

Step 3: Find the probability of both marbles being black.
To find the probability of both marbles being black, we multiply the probabilities of the independent events (drawing a black marble on the first draw and drawing a black marble on the second draw):
P(both black) = P(1st draw = black) * P(2nd draw = black)
P(both black) = (1/6) * (1/6)
P(both black) = 1/36

Therefore, the probability that both marbles drawn are black is 1/36.

(2/12)^2 = ?

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.