1 TAs

We have two 10-hour TAs:

1. Heshan Fernando, fernah, office hour: Wednesday 2pm.

2. Meng Zhang, zhangm19, office hour: Friday 3-4pm.

Their office hours will be on webex, at https://rensselaer.webex.com/meet/RCSID. If no one appears after 15 minutes, they'll leave. OTOH, if you'd like help at other times, contact them. They'll try to be available.

2 Leon Garcia, chapter 2, ctd

1. Leon-Garcia 2.3: Counting methods, pp 41-46.

   1. finite sample space

   2. each outcome equally probable

   3. get some useful formulae

   4. warmup: consider a multiple choice exam where 1st answer has 3 choices, 2nd answer has 5 choices and 3rd answer has 6 choices.

      1. Q: How many ways can a student answer the exam?

      2. A: 3x5x6

   5. If there are k questions, and the i-th question has \(n_i\) answers then the number of possible combinations of answers is \(n_1n_2 .. n_k\)

2. 2.3.1 Sampling WITH replacement and WITH ordering

   1. Consider an urn with n different colored balls.

   2. Repeat k times:

      1. Draw a ball.

      2. Write down its color.

      3. Put it back.

   3. Number of distinct ordered k-tuples = \(n^k\)

3. Example 2.1.5. How many distinct ordered pairs for 2 balls from 5? 5*5.

4. Review. Suppose I want to eat one of the following 4 places, for tonight and again tomorrow, and don't care if I eat at the same place both times: Commons, Sage, Union, Knotty Pine. How many choices to I have where to eat?

   1. 16

   2. 12

   3. 8

   4. 4

   5. something else

5. 2.3.2 Sampling WITHOUT replacement and WITH ordering

   1. Consider an urn with n different colored balls.

   2. Repeat k times:

      1. Draw a ball.

      2. Write down its color.

      3. Don't put it back.

   3. Number of distinct ordered k-tuples = n(n-1)(n-2)...(n-k+1)

6. Review. Suppose I want to visit two of the following four cities: Buffalo, Miami, Boston, New York. I don't want to visit one city twice, and the order matters. How many choices to I have how to visit?

   1. 16

   2. 12

   3. 8

   4. 4

   5. something else

7. Example 2.1.6: Draw 2 balls from 5 w/o replacement.

   1. 5 choices for 1st ball, 4 for 2nd. 20 outcomes.

   2. Probability that 1st ball is larger?

   3. List the 20 outcomes. 10 have 1st ball larger. P=1/2.

8. Example 2.1.7: Draw 3 balls from 5 with replacement. What's the probability they're all different?

   1. P = \(\small \frac{\text{# cases where they're different}}{\text{# cases where I don't care}}\)

   2. P = \(\small \frac{\text{# case w/o replacement}}{\text{# cases w replacement}}\)

   3. P = \(\frac{5*4*3}{5*5*5}\)

9. 2.3.3 Permutations of n distinct objects

   1. Distinct means that you can tell the objects apart.

   2. This is sampling w/o replacement for k=n

   3. 1.2.3.4...n = n!

   4. It grows fast. 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 7!=5040

   5. Stirling approx:

      \begin{equation*} n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n\left(1+\frac{1}{12n}+...\right) \end{equation*}

   6. Therefore if you ignore the last term, the relative error is about 1/(12n).

10. Example 2.1.8. # permutations of 3 objects. 6!

11. Example 2.1.9. 12 airplane crashes last year. Assume independent, uniform, etc, etc. What's probability of exactly one in each month?

    1. For each crash, let the outcome be its month.

    2. Number of events for all 12 crashes = \(12^{12}\)

    3. Number of events for 12 crashes in 12 different months = 12!

    4. Probability = \(12!/(12^{12}) = 0.000054\)

    5. Random does not mean evenly spaced.

12. 2.3.4 Sampling w/o replacement and w/o ordering

    1. We care what objects we pick but not the order

    2. E.g., drawing a hand of cards.

    3. term: Combinations of k objects selected from n. Binomial coefficient.

       \begin{equation*} C^n_k = {n \choose k} = \frac{n!}{k! (n-k)!} \end{equation*}

    4. Permutations is when order matters.

13. Example 2.20. Select 2 from 5 w/o order. \(5\choose 2\)

14. Example 2.21 # permutations of k black and n-k white balls. This is choosing k from n.

15. Example 2.22. 10 of 50 items are bad. What's probability 5 of 10 selected randomly are bad?

    1. # ways to have 10 bad items in 50 is \(50\choose 10\)

    2. # ways to have exactly 5 bad is 3 ways to select 5 good from 40 times # ways to select 5 bad from 10 = \({40\choose5} {10\choose5}\)

    3. Probability is ratio.

16. Multinomial coefficient: Partition n items into sets of size \(k_1, k_2, ... k_j, \sum k_i=n\)

    \begin{equation*} \frac{n!}{k_1! k_2! ... k_j!} \end{equation*}

17. 2.3.5. skip

Reading: 2.4 Conditional probability, page 47-

1. New stuff, pp. 47-66:

   1. Conditional probability - If you know that event A has occurred, does that change the probability that event B has occurred?

   2. Independence of events - If no, then A and B are independent.

   3. Sequential experiments - Find the probability of a sequence of experiments from the probabilities of the separate steps.

   4. Binomial probabilities - tossing a sequence of unfair coins.

   5. Multinomial probabilities - tossing a sequence of unfair dice.

   6. Geometric probabilities - toss a coin until you see the 1st head.

   7. Sequences of dependent experiments - What you see in step 1 influences what you do in step 2.

3 To watch

Rich Radke's Probability Bites:

12. The Monty Hall Problem

13. Independent Events

14. Bernoulli Trials

https://www.youtube.com/playlist?list=PLuh62Q4Sv7BXkeKW4J_2WQBlYhKs_k-pj

4 Occasional no-work classes

Since there is no spring break, or even many long weekends, occasionally the probability class will be something easy, with no homework and no exam questions, like watching a video.

5 Probability in the real world - enrichment

See examples in the random section below.

6 Homework 3

is online here , due in a week.

7 Xkcd comic

Linear Regression