A first box contains 2 black and 2 green pens and a second box contains 4 green and 6 red pens. A random pen is taken from the second box and placed in the first, and then a random pen is taken from the first box and placed in the second. What is the probability that the boxes will return to their original color composition?
P(red,red) = 6/10 * 1/5
P(green,green) = 4/10 * 3/5
so add them up
I don’t get what u are multiplying with . Why two time red, red.
And what happens with those black pens
P(red,red) means there is a 6/10 chance of getting a red pen out of box #2.
Then, since there is now exactly 1 red out of the 5 pens now in box #1, there is a 1/5 chance of getting it. Since the two draws are independent events, you multiply the probabilities.
similarly for the green draws
who cares about the black pens? We are dealing with the chance of getting both draws the same, and the 1st draw cannot be black.
To solve this problem, we need to consider the different possibilities for the color of the pen taken from the second box.
First, let's calculate the probability of each color being chosen from the second box:
- Probability of green pen being chosen from the second box: 4/(4+6) = 4/10 = 2/5.
- Probability of red pen being chosen from the second box: 6/(4+6) = 6/10 = 3/5.
Now, let's consider each possibility for the color of the pen taken from the second box:
1) If a green pen is chosen from the second box, the probabilities for each outcome are as follows:
- Probability of a green pen from the first box being chosen (and returned to the second box): (2+1)/(2+2+1) = 3/5.
- Probability of a black pen from the first box being chosen (and returned to the second box): 2/(2+2+1) = 2/5.
Since there are 2 green pens and 2 black pens in the first box, we have an equal chance of choosing either a green or a black pen.
2) If a red pen is chosen from the second box, the probabilities for each outcome are as follows:
- Probability of a green pen from the first box being chosen (and returned to the second box): 2/(2+2+1) = 2/5.
- Probability of a black pen from the first box being chosen (and returned to the second box): (2+1)/(2+2+1) = 3/5.
Again, we have an equal chance of choosing either a green or a black pen from the first box.
In order for the boxes to return to their original color composition, we need to choose a green pen from the second box and then choose a green pen from the first box. So, we multiply the probabilities of each step:
(2/5) * (3/5) = 6/25
Therefore, the probability that the boxes will return to their original color composition is 6/25.