12.3.3

Section 12.3.3 Blog Entry

Having read section 12.3.3 in the text:

Difficult:
On page 811, I wasn't sure why the text put such emphasis on making sure that both (12.13) and (12.14) must hold in order to show that three events A, B, and C are independent. According to the previous two examples 5 and 6 and the blue definition box above example 5 on page 810, that "two events A and B are independent if: P(A intersection B) = P(A) P(B). I thought that saying "P(A intersection B) = P(A) P(B) P(A intersection C) = P(A) P(C) P(B intersection C) = P(B) P(C)" in (12.13) was the exact same as saying "P(A intersection B intersection C) = P(A) P(B) P(C)" in (12.14). Is there a difference? Is that why the text wants us to make sure we understand that BOTH conditions must be present for the events to be proven independent of one another?

On the top of page 812, I wasn't sure what the text was saying about the number of conditions that followed each of the various numbers of sets: "When there are three sets, we just saw that there are four conditions--namely, there are (3 choose 2) conditions involving pairs of sets and (3 choose 3) conditions involving all three sets. With four sets, there is a total of (4 choose 2) + (4 choose 3) + (4 choose 4) =11; with five sets, 26 conditions; and with ten sets, 1013 conditions." I can see that when there is a pair (events A and B), you only have to check "P(A intersection B) = P(A) P(B)" as the formula states and I can see that the logic also works with three sets, but once the text begins talking about 4 and 5 sets, I can't seem to figure out how they came up with all the 11 combinations, etc. When I tried to think about and write out the conditions for the four sets example, I think I could only come up with about 7 rather than 11.

Reflective:
I think my biggest problem during this section (12.3 in general) is getting used to the notations and determining whether I am showing the correct steps/work for each of the problems. Personally, I feel that I am having a bit of trouble writing down/choosing variables for the probability problems and choosing the proper equations to use. I think thinking first about which equations/formulas/methods to use is a bit confusing for me, and I realized that just reading the assigned math problem and using elementary steps to first figure out what the problem is saying/asking for (tree diagrams, etc.) and then using the probabilities and fractions to get the answer seemed a bit easier to understand in my mind than using the actual formulas. I am hoping that if I practice enough, I will get used to writing down the formulas and applying them to the correct types of problems in the future.