A box contains 4 red cubes, 3 blue cubes, 2 green cubes, and 1 yellow cube. You pick two cubes at random, replacing each after you pick. Find the probability of each compound event.
possible events:
RR RB RG RY
BB BG BY
GG GY
YY
(The order does not matter)
I will do one of them, you do the rest
Prob(B or G) = C(3,1)*C(2,1)/C(10,2) = 6/45 = 2/15
or
Prob(BG) = (3/10)(2/9) = 6/90
prob (GB) = (2/10)(3/9) = 6/90
prob (B or G) = 6/90 + 6/90 = 6/45
To find the probability of compound events, we need to break down the problem into smaller, simpler events. In this case, let's consider the probability of each step separately.
Step 1: Probability of picking a cube
The total number of cubes in the box is 4 + 3 + 2 + 1 = 10. So, there are 10 possible outcomes for this step.
Probability of picking a red cube: There are 4 red cubes, so the probability of selecting a red cube is 4/10.
Probability of picking a blue cube: There are 3 blue cubes, so the probability of selecting a blue cube is 3/10.
Probability of picking a green cube: There are 2 green cubes, so the probability of selecting a green cube is 2/10.
Probability of picking a yellow cube: There is 1 yellow cube, so the probability of selecting a yellow cube is 1/10.
Step 2: Probability of picking a second cube (after replacement)
Since the cubes are replaced after each pick, the probabilities remain the same for the second pick.
Now, let's consider the different compound events you mentioned:
1. Probability of picking two red cubes:
The probability of picking a red cube on the first pick is 4/10. Since we replace the cube before making the second pick, the probability of picking a red cube on the second pick is also 4/10. To find the probability of both events happening, we multiply the probabilities: (4/10) * (4/10) = 16/100 = 4/25.
2. Probability of picking a red cube and then a blue cube:
The probability of picking a red cube on the first pick is 4/10. Since we replace the cube before making the second pick, the probability of picking a blue cube on the second pick is 3/10. To find the probability of both events happening, we multiply the probabilities: (4/10) * (3/10) = 12/100 = 3/25.
3. Probability of picking a green cube twice:
The probability of picking a green cube on the first pick is 2/10. Since we replace the cube before making the second pick, the probability of picking a green cube on the second pick is also 2/10. To find the probability of both events happening, we multiply the probabilities: (2/10) * (2/10) = 4/100 = 1/25.
4. Probability of picking a yellow cube and then any cube:
The probability of picking a yellow cube on the first pick is 1/10. Since we replace the cube before making the second pick, the probability of picking any cube (red, blue, green, or yellow) on the second pick is 10/10 (as there are 10 cubes left in the box). To find the probability of both events happening, we multiply the probabilities: (1/10) * (10/10) = 10/100 = 1/10.
I hope this explanation helps! Let me know if you have any further questions.