Question

a bag contains 4 red, 3 yellow, 3 blue, and 2 white marbles, for a total of 12 marbles. if lucia randomly picks two marbles from the bag without putting the first marble back, what is the probability that the first marble is red and the second is white? 2/144 3/144 2/33 2/12

Explanation:

Step1: Calculate probability of first - red

The probability of picking a red marble first is the number of red marbles divided by the total number of marbles. There are 4 red marbles and 12 total marbles, so $P(\text{first - red})=\frac{4}{12}$.

Step2: Calculate probability of second - white given first - red

After picking a red marble first (without replacement), there are now 11 marbles left in the bag. There are 2 white marbles, so $P(\text{second - white}|\text{first - red})=\frac{2}{11}$.

Step3: Calculate joint probability

By the multiplication rule for conditional probability $P(A\cap B)=P(A)\times P(B|A)$. Here, $A$ is the event of picking a red marble first and $B$ is the event of picking a white marble second. So $P(\text{first - red and second - white})=\frac{4}{12}\times\frac{2}{11}=\frac{8}{132}=\frac{2}{33}$.

Answer:

$\frac{2}{33}$