Conditional Probability and Independence

The ideas of conditional probability and independence were covered today. Using a table, it's relatively straight-forward to demonstrate what conditional probability is. I tried a new demonstration using card guessing and hints to help illustrate independence. Ultimately, I used the definition of P(A | B) = P(A) for independence, then went into the general multiplication rule P(A) x P(B) = P(A | B) x P(B) to show that an equivalent demonstration of independence would be P(A) x P(B) = P(A and B).

These ideas are the easy part. Going into Bayes Theorem is much more confusing. I like to use tree diagrams and work through determining the probability that we are interested in and then focusing on the probability that is of interest out of that total amount. I tell students to not get hung up on the formula but to consider what we are interested in calculating. The tree diagrams help to sort this out and, as probabilities are calculated for the different branches, we ultimately perform the calculations used in Bayes Theorem without getting hung up on what pieces fits where within the formula.

I wrapped up class with the birthday problem. This illustrates how quickly a probability calculation can become messy and for students to think about calculating the compliment of the situation.

Below is an outline of today's lesson along with italicized comments enclosed in square brackets, [like this]

·         Conditional Probability and Independence

o   M and Ms

   §  P(orange| peanut) [referenced back to the original M and M problem to kick things off]

o   Asthma and smoking

   §  Table

   §  Venn Diagram [did not demonstrate this today, will reference it tomorrow]

   §  Tree [used the Three Strings tree diagram to illustrate this]

o   Card probabilities

   §  Ask for volunteer

   §  Choose card from deck; student can win $1 if guesses drawn card (no suit necessary); student writes guess on slip of paper which is not shared with anyone

   §  Ask class “what is the probability of having a correct guess?” answer 4/52

   §  For a penny give a hint—card is red (or black)

   §  “what is the probability of a correct guess?” answer 2/26

   §  For another penny another hint—card is a heart (or appropriate suit)

   §  “what is the probability of a correct guess?” answer 1/13

   §  For another penny give one more hint—card is a number (or face card)

   §  “what is the probability of having a correct guess?” answer 1/9 or ¼

   §  Discuss what information was helpful and what wasn’t

   §  Talk about independence informally [this went as described, really had students think about what information influenced any change in guess]

o   Independence

   §  Go back through problems to check on independence

      ·         M and Ms [asked if being orange and type were independent, had students demonstrate using both definitions]

      ·         Roulette [didn't use]

      ·         Asthma [students looked at whether living with a smoker and having asthma were independent]

      ·         Clothing and gender [didn't use]

o   Bayes

   §  P(plain | orange) [students calculated and then showed tree diagram]

   §  Asthma [didn't use, looked at medical testing results problem from book]

   §  Clothing (shorts or jeans, etc depending on what you see in class) and gender [didn't use, may do this next class]

·         Counting and probability
o   The Birthday problem

   §  Have students record day of birth in month column [turned out we had four pairs of same birthday, one of which had three individuals]

   §  Pose question [the question being is it unusual to have a matched birthday or is it something we should expect]

   §  Results involve factorials and permutations [after students worked on the problem a bit, discussed the direct calculation, building from two students to three to four; then talked about calculating no matching birthdays, where the result is much easier to calculate; this was homework, along with working  through some Bayes Theorem problems]