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Engineering Probability Class 3 Mon 2021-02-01

Source

1 Homework 2

is online, due in a week.

3 Chapter 2 ctd

  1. Corollory 6:

    \(\begin{array}{c} P\left[\cup_{i=1}^n A_i\right] = \\ \sum_{i=1}^n P[A_i] \\ - \sum_{i<j} P[A_i\cap A_j] \\ + \sum_{i<j<k} P[A_i\cap A_j\cap A_k] \cdots \\ + (-1)^{n+1} P[\cap_{i=1}^n A_i] \end{array}\)

    1. Example Q=queen card, H=heart, F= face card.

      1. P[Q]=4/52, P[H]=13/52, P[F]=12/52,

      2. P[Q \(\cap\) H]=1/52, P[Q \(\cap\) F] = ''you tell me''

      3. P[H \(\cap\) F]= ''you tell me''

      4. P[Q \(\cap\) H \(\cap\) F] = ''you tell me''

      5. So P[Q \(\cup\) H \(\cup\) F] = ?

    2. Example from Roulette:

      1. R=red, B=black, E=even, A=1-12

      2. P[R] = P[B] = P[E] = 16/38. P[A]=12/38

      3. \(P[R\cup E \cup A]\) = ?

  2. Corollory 7: if \(A\subset B\) then P[A] <= P[B]

    Example: Probability of a repeated coin toss having its first head in the 2nd-4th toss (1/2+1/4+1/8) \(\ge\) Probability of it happening in the 3rd toss (1/4).

  3. 2.2.1 Discrete sample space

    1. If sample space is finite, probabilities of all the outcomes tell you everything.

    2. sometimes they're all equal.

    3. Then P[event]} \(= \frac{\text{#. outcomes in event}}{\text{total # outcomes}}\)

    4. For countably infinite sample space, probabilities of all the outcomes also tell you everything.

    5. E.g. fair coin. P[even] = 1/2

    6. E.g. example 2.9. Try numbers from random.org.

    7. What probabilities to assign to outcomes is a good question.

    8. Example 2.10. Toss coin 3 times.

      1. Choice 1: outcomes are TTT ... HHH, each with probability 1/8

      2. Choice 2: outcomes are # heads: 0...3, each with probability 1/4.

      3. Incompatible. What are probabilities of # heads for choice 1?

      4. Which is correct?

      5. Both might be mathematically ok.

      6. It depends on what physical system you are modeling.

      7. You might try doing the experiment and observing.

      8. You might add a new assumption: The coin is fair and the tosses independent.

  4. Example 2.11: countably infinite sample space.

    1. Toss fair coin, outcome is # tosses until 1st head.

    2. What are reasonable probabilities?

    3. Do they sum to 1?

  5. 2.2.2 Continuous sample spaces

    1. Usually we can't assign probabilities to points on real line. (It just doesn't work out mathematically.)

    2. Work with set of intervals, and Boolean operations on them.

    3. Set may be finite or countable.

    4. This set of events is a ''Borel set''.

    5. Notation:

      1. [a,b] closed. includes both. a<=x<=b

      2. (a,b) open. includes neither. a<x<b

      3. [a,b) includes a but not b, a<=x<b

      4. (a,b] includes b but not a, a<x<=b

    6. Assign probabilities to intervals (open or closed).

    7. E.g., uniform distribution on [0,1] \(P[a\le x\le b] = \frac{1}{b-a}\)

    8. Nonuniform distributions are common.

    9. Even with a continuous sample space, a few specific points might have probabilities. The following is mathematically a valid probability distribution. However I can't immediately think of a physical system that it models.

      1. \(S = \{ x | 0\le x\le 1 \}\)

      2. \(p(x=1) = 1/2\)

      3. For \(0\le x_0 \le 1, p(x<x_0) = x_0/2\)

  6. For fun: Heads you win, tails... you win. You can beat the toss of a coin and here's how....

  7. Example 2.13, page 39, nonuniform distribution: chip lifetime.

    1. Propose that P[(t, \(\infty\) )] = \(e^{-at}\) for t>0.

    2. Does this satisfy the axioms?

    3. I: yes >0

    4. II: yes, P[S] = \(e^0\) = 1

    5. III here is more like a definition for the probability of a finite interval

    6. P[(r,s)] = P[(r, \(\infty\) )] - P[(s, \(\infty\) )] = \(e^{-ar} - e^{-as}\)

  8. Probability of a precise value occurring is 0, but it still can occur, since SOME value has to occur.

  9. Example 2.14: picking 2 numbers randomly in a unit square.

    1. Assume that the probability of a point falling in a particular region is proportional to the area of that region.

    2. E.g. P[x>1/2 and y<1/10] = 1/20

    3. P[x>y] = 1/2

  10. Recap:

    1. Problem statement defines a random experiment

    2. with an experimental procedure and set of measurements and observations

    3. that determine the possible outcomes and sample space

    4. Make an initial probability assignment

    5. based on experience or whatever

    6. that satisfies the axioms.

4 To watch

Rich Radke's Probability Bites:

  1. Combinatorics Practice Problems

  2. Continuous Sample Spaces

  3. Conditional Probability

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