Having a hard time with this question: Help Please
This is a multi-stage experiment. There are 7 colored cubes in a bag. The colors are purple, blue, red, orange, white, yellow, and green. This is a multi-stage experiment with dependent events. Determine the probabilities of the following events. Express probabilities as fractions.
P(OWB)=
P(PRBO)=
P(GBOWR)=
P(GRBGW)=
P(PBRGOY)=
Is this with replacement, or without?
I think it is with replacement. The question doesn't say
To determine the probabilities of these events, we need to understand the concept of dependent events.
Dependent events are events where the outcome of one event affects the outcome of subsequent events. In this case, the events are selecting colored cubes from the bag without replacement, meaning that once a cube is selected, it is not put back into the bag.
Let's break down the given events and calculate the probabilities step by step:
Event P(OWB):
1. Determine the probability of selecting an orange cube first: P(O) = Number of orange cubes / Total number of cubes
P(O) = 1/7
2. Determine the probability of selecting a white cube after selecting an orange cube: P(W) = Number of white cubes / Number of remaining cubes
P(W) = 1/6
3. Determine the probability of selecting a blue cube after selecting an orange and a white cube: P(B) = Number of blue cubes / Number of remaining cubes
P(B) = 1/5
4. Multiply the probabilities together to get the final probability:
P(OWB) = P(O) * P(W) * P(B)
Event P(PRBO):
1. Determine the probability of selecting a purple cube first: P(P) = Number of purple cubes / Total number of cubes
P(P) = 1/7
2. Determine the probability of selecting a red cube after selecting a purple cube: P(R) = Number of red cubes / Number of remaining cubes
P(R) = 1/6
3. Determine the probability of selecting a blue cube after selecting a purple and a red cube: P(B) = Number of blue cubes / Number of remaining cubes
P(B) = 1/5
4. Determine the probability of selecting an orange cube after selecting a purple, red, and blue cube: P(O) = Number of orange cubes / Number of remaining cubes
P(O) = 1/4
5. Multiply the probabilities together to get the final probability:
P(PRBO) = P(P) * P(R) * P(B) * P(O)
Similarly, you can calculate the probabilities for events P(GBOWR) and P(GRBGW) following the steps outlined above.
For the last event, P(PBRGOY), you need to determine the probability of selecting a purple cube, followed by a blue cube, a red cube, a green cube, an orange cube, and finally, a yellow cube. You can apply the same steps as before to calculate the probability of each individual step and then multiply them together to get the final probability.
Remember to express the probabilities as fractions by reducing them to their simplest form.
I hope this explanation helps you solve the problem. Let me know if you have any further questions!