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?
To find the conditional probability, we need to use the definition of conditional probability. The conditional probability of an event (B) given another event (A) is equal to the probability of both events happening together (A and B) divided by the probability of the first event (A) occurring.
In this case, we want to find the probability of drawing a purple marble (event B) given that the first marble drawn is green (event A).
Let's break down the problem step by step:
Step 1: Determine the probability of drawing a green marble.
There are a total of 2 blue, 3 purple, 4 yellow, 2 red, 3 green, and 1 orange marbles in the bag. Therefore, the probability of drawing a green marble as the first marble is 3/17 (there are 3 green marbles out of 17 total marbles).
Step 2: Determine the probability of drawing a purple marble after drawing a green marble.
Now, we have one green marble remaining in the bag and a total of 16 marbles left. The number of remaining purple marbles is 3, and the total number of marbles left is 16. So, the probability of drawing a purple marble after drawing a green marble is 3/16.
Step 3: Calculate the conditional probability.
Now we can plug the values into the formula:
Conditional probability (purple given green) = Probability of (green and purple) / Probability of green
= (3/16) / (3/17)
= (3/16) * (17/3)
= 1/16
Therefore, the conditional probability that the second marble drawn is purple given that the first marble is green is 1/16.