A draw contains 3 black socks and 2 white socks. A sock isdrawn at random and then replaced. Find each probability. p(2 black) p(black, then white) p(white, then black) p(2white)

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

P(2black) = 3/5 * 3/5 = 9/15 = 3/5 = .6

Use the same process for the remaining probabilities.

To find each probability, we need to understand the concept of probability and the rules of probability.

Probability is the likelihood of an event occurring, and it is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Let's calculate each probability step by step:

1. P(2 black):
To calculate the probability of drawing 2 black socks, we need to find the probability of drawing a black sock on the first draw and then replacing it, and then drawing another black sock.
The first draw:
The probability of drawing a black sock on the first draw is 3 black socks / 5 total socks (3 black + 2 white) since there are 3 black socks out of 5 total socks.

Since the sock is replaced after the first draw, the second draw is independent of the first draw.
The second draw:
The probability of drawing another black sock on the second draw is also 3 black socks / 5 total socks since we replaced the first sock.

To find the probability of both events happening, we multiply the probabilities of each event:
P(2 black) = (3/5) * (3/5) = 9/25

Therefore, the probability of drawing two black socks is 9/25.

2. P(black, then white):
To calculate the probability of drawing a black sock and then a white sock, we need to find the probability of drawing a black sock on the first draw (as explained above) and, without replacing the black sock, drawing a white sock on the second draw.

First draw:
The probability of drawing a black sock on the first draw is still 3/5.

Second draw:
Since we didn't replace the first black sock, there would be 4 socks left for the second draw (3 black + 1 white). So, the probability of drawing a white sock on the second draw is 1 white sock / 4 total socks.

To find the overall probability, we multiply the probabilities of each event:
P(black, then white) = (3/5) * (1/4) = 3/20

Therefore, the probability of drawing a black sock and then a white sock is 3/20.

3. P(white, then black):
To calculate the probability of drawing a white sock and then a black sock, we need to find the probability of drawing a white sock on the first draw and then a black sock on the second draw.

First draw:
The probability of drawing a white sock on the first draw is 2 white socks / 5 total socks since there are 2 white socks out of 5 total socks.

Second draw:
Since we replaced the first white sock, there would still be 5 socks for the second draw (3 black + 2 white). So, the probability of drawing a black sock on the second draw is 3 black socks / 5 total socks.

To find the overall probability, we multiply the probabilities of each event:
P(white, then black) = (2/5) * (3/5) = 6/25

Therefore, the probability of drawing a white sock and then a black sock is 6/25.

4. P(2 white):
To calculate the probability of drawing 2 white socks, we need to find the probability of drawing a white sock on the first draw (as explained above) and, after replacing the sock, drawing another white sock.

The first draw:
The probability of drawing a white sock on the first draw is 2 white socks / 5 total socks.

Since the sock is replaced after the first draw, the second draw is independent of the first draw.
The second draw:
The probability of drawing another white sock on the second draw is also 2 white socks / 5 total socks since we replaced the first sock.

To find the overall probability, we multiply the probabilities of each event:
P(2 white) = (2/5) * (2/5) = 4/25

Therefore, the probability of drawing two white socks is 4/25.

In summary:
P(2 black) = 9/25
P(black, then white) = 3/20
P(white, then black) = 6/25
P(2 white) = 4/25

To find each probability, we need to determine the total number of socks, as well as the number of each type of sock. Let's calculate each probability step by step:

Step 1: Calculate the total number of socks:
The draw contains 3 black socks and 2 white socks, so the total number of socks is 3 + 2 = 5.

Step 2: Calculate p(2 black):
For this probability, we need to find the chance of drawing 2 black socks. Since we replace the sock after each draw, the probability of drawing a black sock in any single draw is 3/5. To find the probability of drawing 2 black socks, we multiply this probability by itself:

p(2 black) = (3/5) * (3/5) = 9/25.

Step 3: Calculate p(black, then white):
To calculate this probability, we have to find the chance of drawing a black sock first and then a white sock second. Since the socks are replaced after each draw, the probability of drawing a black sock on the first draw is 3/5, and the probability of drawing a white sock on the second draw is 2/5. Multiplying these probabilities together gives us:

p(black, then white) = (3/5) * (2/5) = 6/25.

Step 4: Calculate p(white, then black):
Similar to the previous step, we need to find the probability of drawing a white sock first and then a black sock second. Since the socks are replaced after each draw, the probability of drawing a white sock on the first draw is 2/5, and the probability of drawing a black sock on the second draw is 3/5. Multiplying these probabilities together gives us:

p(white, then black) = (2/5) * (3/5) = 6/25.

Step 5: Calculate p(2 white):
For this probability, we need to find the chance of drawing 2 white socks. Since we replace the sock after each draw, the probability of drawing a white sock in any single draw is 2/5. To find the probability of drawing 2 white socks, we multiply this probability by itself:

p(2 white) = (2/5) * (2/5) = 4/25.

So, the probabilities are as follows:
p(2 black) = 9/25
p(black, then white) = 6/25
p(white, then black) = 6/25
p(2 white) = 4/25