Given the following probability formula: P(A and B) = P(A) x P(B), can you provide a different example where two events are selected at random and you need to find the probability of both events occurring? Please show your work.
Sure! Let's consider an example where you have a bag containing 10 marbles: 4 red marbles and 6 blue marbles. You randomly select two marbles from the bag, without replacement.
Now, we want to find the probability of selecting a red marble on the first draw AND selecting a blue marble on the second draw.
To calculate this, we'll use the formula P(A and B) = P(A) x P(B), where A represents the event of selecting a red marble on the first draw and B represents the event of selecting a blue marble on the second draw.
First, let's find the probability of selecting a red marble on the first draw (P(A)):
P(A) = (number of red marbles) / (total number of marbles)
P(A) = 4 / 10
P(A) = 2 / 5
Now, since we didn't replace the first marble, there are now 9 marbles left in the bag, with 3 red marbles and 6 blue marbles. Thus, we'll find the probability of selecting a blue marble on the second draw (P(B)):
P(B) = (number of blue marbles) / (total number of marbles left)
P(B) = 6 / 9
P(B) = 2 / 3
Next, we'll calculate the probability of both events occurring by multiplying the individual probabilities:
P(A and B) = P(A) x P(B)
P(A and B) = (2 / 5) x (2 / 3)
P(A and B) = 4 / 15
So, the probability of selecting a red marble on the first draw AND selecting a blue marble on the second draw is 4/15.