Since Jared had to leave class early, I thought it best to continue with our Algebra topic rather than switch to a new subject.

We continued studying conditional probabilities, both in table format and in formula format. Jared worked through some examples to familiarize himself with the formula. We then started exploring probability trees, where each branch represents a fractional probability. The sum of the branches equals 1, representing the full sample space of probabilities.

From probability distributions we moved onto exploring conditional probabilities. Conditional probabilities contain the condition, as the name implies, that one event occurs provided that another event has already occurred, or 'given the probability of event, given event A'. This can include situations involving either/or, such as whether a type of material is recycled or not, or whether household chores are undertaken depending on gender of the family member involved. Jared studied a few practical examples where the information was laid out in table format. We finished class by expressing this as a formula, i.e. the probability of event B to happen given event A = P ( A and B ) / P ( A ).

Jared worked on some problems identifying what information to extract from a table so as to present the data in graphic format, for example. Probability distributions can be represented in bar graph format, or even in circle graph format. We also looked at how to convert odds into probabilities. It's important to identify what the total outcome is when odds in favor or against are expressed. Odds in favor, say, expressed as 1:4 implies a 20% probability that the outcome will be in favor, not a 25% probability, because the total number of outcomes is 5, not 4.

Jared has worked quite hard this week, and I thought it would be nice for him to learn a bit about chaos theory. He has expressed an interest in this subject in the past, and I when I asked him to look up a documentary he found an excellent one produced by the BBC. The first part of the documentary covered the early history of chaos theory, primarily championed by Henri Poincare in France and Aleksandr Lyapunov in Russia (although it was not known by that name yet). The former studied the effect of small-scale perturbations in the initial conditions of orbiting bodies (such as the planets around the Sun), while the latter studied the chaotic behavior of gases and fluids.

Today we reviewed the terminology and some initial steps required to start investigating Probability Distributions. One of the first things one can do when arranging data is to draw up a 'frequency table' that lists the number of times outcomes occur in a sample space. From this information, one can easily calculate probabilities of the different outcomes, and arrange the data as a bar graph, for example. A 'cumulative probability' describes a probability over a continuous range of events, such as the number of times something occurs over a week.

It was so very exciting to get Jared's news re Eckerd College and USF! We spent much of class talking about the colleges, and university programs, and all that. It was hard to get back on topic (I was so thrilled for Jared), but we recapped a bit on what we had looked at last week, on combining probabilities.

Since this is the second day that Jared is at home, I decided not to lose more time and so we looked at some Statistics. There were two problems on the test Jared took at the end of last quarter that needed revision. These included the concept of calculating the probability in an 'at least one' situation, for which the complement i.e. 'none' is used. We also covered Combinations again, and embarked on using combinations to calculate probabilities for certain situations, such as the probability of being dealt a 5-card flush in a hand of poker from a standard deck of 52 cards.

We worked on Algebra during this class (please see the earlier notes) - I find it more productive to work on algebra when Jared is on Skype. We always catch up on statistics when he is present at the school. An excellent two hours today, despite Jared's sore throat and sniffly nose,

We finally watched the end of the documentary that has gripped Jared since Friday. The last ten minutes or so dealt with rather philosophical topics, such as the question whether math is inherent to humans (we tend to understand the world around us naturally through numbers, for example we have two hands, five toes, etc, and therefore we understand the concept of '2' and '5') or whether it is intrinsic to the universe, and we developed a 'language' to understand it.