Suppose a jar contains 5 red marbles and 14 blue marbles. If you reach in the jar and pull out 2 marbles at random at the same time, find the probability that both are red.
To find the probability that both marbles are red, we can use the formula for the probability of independent events:
P(A and B) = P(A) * P(B|A)
Here, event A is pulling out a red marble on the first draw, and event B is pulling out a red marble on the second draw, given that we already pulled out a red marble on the first draw.
First, let's find the probability of event A:
There are 5 red marbles and a total of 5 + 14 = 19 marbles in the jar. So the probability of drawing a red marble on the first draw is:
P(A) = 5/19
Now let's find the probability of event B, given that we already drew a red marble on the first draw:
Since we already drew one red marble, there are now 4 red marbles left in the jar, and a total of 4 + 14 = 18 marbles.
P(B|A) = 4/18
Now we can find the probability of both events A and B occurring:
P(A and B) = P(A) * P(B|A)
P(A and B) = (5/19) * (4/18)
P(A and B) = 20/342
Simplify the fraction:
P(A and B) = 10/171
So the probability that both marbles are red is 10/171.