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 is green?
if the first is green, there are still 14 marbles left, and the purples are still all there, so
prob = 3/14
To find the conditional probability that the second marble is purple given that the first marble is green, we need to use the formula for conditional probability:
P(A|B) = P(A and B) / P(B)
In this case, A represents the event of drawing a purple marble on the second draw, and B represents the event of drawing a green marble on the first draw.
To calculate P(A and B), we need to consider two things:
1. The probability of drawing a green marble on the first draw: P(B) = (number of green marbles) / (total number of marbles)
2. The probability of drawing a purple marble on the second draw, given that a green marble was already drawn: P(A and B) = (number of purple marbles after a green marble is drawn) / (total number of marbles left after the green marble is drawn)
Let's calculate these probabilities step by step:
Step 1: Calculate the probability of drawing a green marble on the first draw.
There are a total of 2 blue, 3 purple, 4 yellow, 2 red, 3 green, and 1 orange marbles. Therefore, the probability of drawing a green marble on the first draw is:
P(B) = (number of green marbles) / (total number of marbles)
= 3 / (2+3+4+2+3+1)
= 3 / 15
= 1 / 5
Step 2: Calculate the probability of drawing a purple marble on the second draw, given that a green marble was already drawn.
After drawing a green marble on the first draw, there are 2 blue, 3 purple, 4 yellow, 2 red, and 1 orange marbles left. Therefore, the probability of drawing a purple marble on the second draw, given that a green marble was already drawn, is:
P(A and B) = (number of purple marbles after a green marble is drawn) / (total number of marbles left after the green marble is drawn)
= 3 / (2+3+4+2+1)
= 3 / 12
= 1 / 4
Now we can calculate the conditional probability using the formula:
P(A|B) = P(A and B) / P(B)
= (1 / 4) / (1 / 5)
= (1 / 4) * (5 / 1)
= 5 / 4
Therefore, the conditional probability that the second marble is purple given that the first marble is green is 5/4.