A bag contains two blue, three purple, four yellow, two red, three green and one orange marbles.
What is the conditional probability that, without replacement, the second marble is purple given that the first marble was green?
3/14 for just the purple.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
3/15 * 3/14 = ? for both events
To find the conditional probability that the second marble is purple given that the first marble was green, we need to use the concept of conditional probability.
Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, we want to find the probability of drawing a purple marble as the second draw, knowing that a green marble was drawn on the first draw.
The formula for conditional probability is:
P(A|B) = P(A ∩ B) / P(B)
Where P(A|B) represents the probability of event A occurring given that event B has already occurred, P(A ∩ B) represents the probability of both events A and B occurring together, and P(B) represents the probability of event B occurring.
In this case, event A is drawing a purple marble on the second draw, and event B is drawing a green marble on the first draw.
First, let's calculate the probability of drawing a green marble on the first draw. There are a total of 2 green marbles and 18 marbles in total, so the probability is:
P(B) = 2/18
Next, let's calculate the probability of drawing a green marble on the first draw and a purple marble on the second draw. After drawing a green marble, we are left with 2 green, 3 purple, 3 yellow, 2 red, and 1 orange marble. So the probability is:
P(A ∩ B) = (3/17) * (2/18)
(The probability of drawing a purple marble on the second draw is 3/17 since there are 3 purple marbles remaining out of the 17 marbles remaining after the green marble was drawn.)
Finally, let's calculate the conditional probability:
P(A|B) = P(A ∩ B) / P(B) = [(3/17) * (2/18)] / (2/18) = 3/17
Therefore, the conditional probability that the second marble is purple given that the first marble was green is 3/17.