Probabilties math

Question

Conditional Probability
Jamal has 12 marbles as shown in the diagram below. Each marble is either blue or white. In addition, each marble is labeled with a number between 1 and 5.
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Jamal chooses one of the marbles at random. Determine the following probabilities. Enter your probabilites as fractions and simplify any fractions as needed.
What is the probability that Jamal will choose a white marble given that the marble is even?
What is the probability that Jamal will choose a even marble given that the marble is blue?

Answer 1

Hi Tra'von,

This is Bayes' Theorem:

P(A given B)=P(A and B)/P(B)

where A and B are independent events. So, to part a.

(a) Designate events as follows:

A=Jamal chooses a white marble

B=Marble is even

Now, the multiplication rule for independent events states:

P(A and B)=P(A)*P(B)

P(A)=1/2 We don't know anything specific about how many blues or whites, so we have to assume it's a 50/50 chance.

P(B)=3/12 Only two marbles of the twelve are even. In reduced form:

P(B)=1/4

P(A and B)=(1/2)*(1/4)=1/8

Now, we want probability of a white given that marble is even.

P(A given B)= P(A and B)/P(B)

P(A given B)=(1/8)/(1/4)

P(A given B)= 1/2

b. Same formula, different event designations.

A=Marble is even

B=Marble is blue

P(A)=3/12=1/4--Same as above, only 3 even marbles in sample of 12

P(B)=1/2--We were not told any difference in number of blue vs. white marbles.

P(A given B)=P(A and B)/P(B)

P(A and B)=P(A)*P(B)

P(A and B)=(1/4)*(1/2)

P(A and B)=1/8

P(A given B)=(1/8)/(1/2)

P(A given B)=1/4=0.25

I hope this helps.