A fair coin is flipped. If the flip results in a head, then a marble is selected from an urn containing 1 red, 4 white, and 9 blue marbles. If the flip results in a tail then a marble is selected from an urn containing 3 red and 3 white marbles. If the marble selected is white, then what is the probability that a flip resulted in a head?
To find the probability that a flip resulted in a head given that a white marble was selected, we can use Bayes' theorem.
Bayes' theorem states that the conditional probability of event A given event B is equal to the probability of event A and B occurring together divided by the probability of event B. We can apply this theorem to our scenario.
Let's denote the events as follows:
H = Flip results in a head
T = Flip results in a tail
W = White marble is selected
We want to find the probability of H given W, which can be written as P(H|W).
According to Bayes' theorem:
P(H|W) = P(W|H) * P(H) / P(W)
Now let's calculate the values needed for the equation:
P(W|H) = Probability of selecting a white marble given that the flip resulted in a head.
From the urn with 1 red, 4 white, and 9 blue marbles, the probability of selecting a white marble is 4/(1+4+9) = 4/14 = 2/7.
P(H) = Probability that the flip resulted in a head.
Since the coin is fair, the probability of a head is 1/2.
P(W) = Probability of selecting a white marble.
To calculate this, we need to consider the total number of marbles selected.
If the flip resulted in a head, there are a total of 14 marbles (1 red, 4 white, and 9 blue).
If the flip resulted in a tail, there are a total of 6 marbles (3 red and 3 white).
So, P(W) = P(W|H) * P(H) + P(W|T) * P(T) = (2/7) * (1/2) + (3/6) * (1/2) = 1/7 + 1/4 = 11/28.
Now we can substitute the values into the equation:
P(H|W) = (2/7) * (1/2) / (11/28)
Simplifying, we get:
P(H|W) = (2/7) * (1/2) * (28/11) = 4/11.
Therefore, the probability that a flip resulted in a head given that a white marble was selected is 4/11.