Dear Sir or Madam,

I am trying to work out the below problem to see how P(A) arrived at 1/2, P(B) arrived at 2/3 and P(A n B) at 1/3. I reviewed and following the previous examples, but this one seem to be a little more complex.

Example 3.19 Checking for IndependenceDieTossing Experiment Problem Consider the experiment of tossing a fair die and let A: {Observe an even number.} B: {Observe a number less than or equal to 4.} Are events A and B independent? Solution The Venn diagram for this experiment is shown in Figure 3.20 IE'. We first calculate PM) = P(2]+P(4)+P(6):% MB) = P(1J+P(2)+P(3)+p(4): PLANE) = P(2]+P(4) :% S Figure 3.20 Venn diagram for die-toss experiment Now assuming B has occurred, the conditional probability ofA given B is P(Am3) _ PU?) _ =1=P(A) P (A|B) = 2 w|m|w|H Thus, assuming that event B occurring does not alter the probability of observing an even number, PM) remains %. Therefore, the events A and B are independent. Look Back Note that if we calculate the conditional probability of B given A, our conclusion is the same