Well, I did it. One full year of KA under my belt. I’ve been doing a lot of reflecting this week (which is saying something since I tend to reflect a lot as it is) and am quite happy with what I’ve been able to learn over the past year. I’m proud of what I’ve accomplished but, looking back on everything I’ve done, I can’t help but think about how much more I need/want to learn. Considering that my progression rate has slowed down substantially working through these more difficult courses, I think getting through the remaining courses of KA will most likely take quite a long time. After getting through the statistics courses, the only remaining subject is calculus, which there are quite a few courses within that subject. My goal is to get through all of the calculus courses by Week 104. First thing’s first, however, I still need to get through statistics.

I began Week 52 in the middle of the section Combining Random Variables. Going through the remainder of this section, I got a lot of practice adding/subtracting normal distributions together and converting their respective standard deviations to variances and adding them together. Here’s a photo from my notes of an example question:

• Q. Adam and Michael like to bowl together. Adam has an average score of 175 with a standard deviation of 30 and Michael has an average score of 150 and a standard deviation of 40. What’s the probability that Michael will have a higher score than Adam the next time they bowl?

• In order to solve this question, you must subtract the means of both their scores to come up with the mean-difference, then convert each players S.D. to variance, add the two S.D.s together and find the sum’s square root to get the S.D.-difference.
  • Mean-Difference = μD =
    • μD = μA – μM
      • = 175 – 150
      • μD = 25
  • σ-Difference = √ (σ_A)^2 + (σ_M)^2
    • = √ (30)^2 + (40)^2
    • = √ 900 + 1600
    • = √ 2500
    • σ-D = 50
• To determine whether Michael will beat Adam, it’s easiest to create another normal distribution with μD = 25 and σ-D = 50 which I did in the bottom right corner of the photo. The part that I don’t understand is that apparently when X = 0 the two players tie, when X is negative Michael wins, and when X is positive Adam wins. Based off their average scores and the fact that there are more positive values on this normal distribution than there are negative values, it makes sense that the negative values represent Michael’s chance of winning and the positive values represent Adam’s chance of winning, however I don’t understand exactly how or why this works.

I then started the next section of the unit labelled Binomial Random Variables. Here I learned that for a variable to be considered binomial it must have four characteristics (apart from the one exception I explain below):

1. Must be made up of independent trials.
2. Each trial can be considered either a ‘success’ or ‘failure’ in a binary sense.
3. There must be a fixed number of trials.
4. The probability of success must remain constant on each trial.

The exception to these four characteristics that determine if a variable is binomial is what’s known as the ‘10% Rule’:

• When running an experiment, if a sample is taken of a population that is greater than 10% of the population (ex. surveying 200/1000 people – 20%) you can assume that each person (i.e. trials within the random variable) is not necessarily independent of each other. However, if the sample is less-than-or-equal-to 10% of the population (ex. 200/10000 – 2%), assuming that the surveyors were picked at random, statisticians will most often infer that the trials (i.e. people) within the sample are independent and therefore the random variable as a whole is binomial.

• The above photo shows how taking a random sample of 3 students out of 20, 30, and 10,000 students changes the samples percentage relative to the population and therefore the variables status as either binomial or not.
• Lastly, it’s worth noting that the graph of a binomial variable which has a 50-50% chance of being either a ‘success’ or ‘failure’ looks very similar a normal distribution. Sal made it clear that they are not the exact same thing, but he also said that if a 50-50% binomial variable is run through 999,999,999,999 (etc.) trials it will essentially look indistinguishable from a normal distribution when viewing the entire distribution from end-to-end.

This section finished by looking at how to calculate a binomial variable when the probability of ‘success’ and ‘failure’ are unequally weighted. Here is a page from my notes that shows the formula and labels its’ constituents:

Here is an example question of how to us the formula:

• Q. Lebron James makes his free throws 70% of the time. If James takes 6 free throws, what’s the probability of him scoring 0, 1, 2, 3, 4, 5, or all 6 shots?

As you can see from the bottom of the page, the weighted distribution is skewed to the left which makes sense since James has a 70% chance of making his free throws.

I heard Sal use the word combinatorics sometime this week and realized that’s the name for the branch of math that deals with combinations and, I assume, permutations. When I heard him use the word combinatorics, he also phrased a question in a way I hadn’t heard before which made the combination notation more clear to me. In the example I heard him talk about, he was flipping a fair coin 5 times and trying to figure out the probability of the coin landing on heads 0 times, i.e. _5(C)_0. The way he phrased it was, “you have 5 flips and you’re choosing 0 of them to be heads.” For some reason that phrasing helped me to wrap my head around the notation. This means, for example, that if you’re trying to figure out the probability of a fair coin landing on heads 8 time of 11 flips, ie. _11(C)_8, you can say, “what’s the probability of flipping a coin 11 times and choosing 8 of the flips to be heads?”.

I have this coming week off from work and I’ve decided I will 100% get through this unit which, in fairness, should be fairly simply. I have 15 videos left to watch and 7 exercises to get through, plus the unit test. I’m also 69% of the way through the entire course so hopefully I’ll be able to wrap it up nearing the end of September or early October. As I’ve mentioned before, I’ve already completed a fair amount of both the other units so, again, I’m really hoping I can get through both of those units by the end of October to start calculus in November.

Lastly, it looks like I’ll be able to tutor at the club I work at this fall! I spoke with the manager who’s in charge of tutoring and said it will likely work out and that she’d be happy to let me give it a try. I went online to the Nelson textbook website and found quizzes from each chapter of their Grade 9 textbook and was happy to find out I was able to do >~95% of the questions without difficulty. I still have a few more quizzes to get through but am fairly confident I’ll know the material. Tutoring feels like a small but important step in a new direction for me so I’m fairly excited (but nervous!) for this opportunity!