Engineering Probability Class 3 Thu 2020-01-23
Table of contents::
1 Homework 2
is online, due next Thurs.
2 Probability in the real world - enrichment
How MIT Students Won $8 Million in the Massachusetts Lottery.
3 Chapter 2 ctd
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Corollory 6:
\(\begin{array}{c} P\left[\cup_{i=1}^n A_i\right] = \\ \sum_{i=1}^n P[A_i] \\ - \sum_{i<j} P[A_i\cap A_j] \\ + \sum_{i<j<k} P[A_i\cap A_j\cap A_k] \cdots \\ + (-1)^{n+1} P[\cap_{i=1}^n A_i] \end{array}\)
- Example Q=queen card, H=heart, F= face card.
- P[Q]=4/52, P[H]=13/52, P[F]=12/52,
- P[Q \(\cap\) H]=1/52, P[Q \(\cap\) F] = ''you tell me''
- P[H \(\cap\) F]= ''you tell me''
- P[Q \(\cap\) H \(\cap\) F] = ''you tell me''
- So P[Q \(\cup\) H \(\cup\) F] = ?
- Example from Roulette:
- R=red, B=black, E=even, A=1-12
- P[R] = P[B] = P[E] = 16/38. P[A]=12/38
- \(P[R\cup E \cup A]\) = ?
- Example Q=queen card, H=heart, F= face card.
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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).
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2.2.1 Discrete sample space
- If sample space is finite, probabilities of all the outcomes tell you everything.
- sometimes they're all equal.
- Then P[event]} \(= \frac{\text{#. outcomes in event}}{\text{total # outcomes}}\)
- For countably infinite sample space, probabilities of all the outcomes also tell you everything.
- E.g. fair coin. P[even] = 1/2
- E.g. example 2.9. Try numbers from random.org.
- What probabilities to assign to outcomes is a good question.
- Example 2.10. Toss coin 3 times.
- Choice 1: outcomes are TTT ... HHH, each with probability 1/8
- Choice 2: outcomes are # heads: 0...3, each with probability 1/4.
- Incompatible. What are probabilities of # heads for choice 1?
- Which is correct?
- Both might be mathematically ok.
- It depends on what physical system you are modeling.
- You might try doing the experiment and observing.
- You might add a new assumption: The coin is fair and the tosses independent.
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Example 2.11: countably infinite sample space.
- Toss fair coin, outcome is # tosses until 1st head.
- What are reasonable probabilities?
- Do they sum to 1?
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2.2.2 Continuous sample spaces
- Usually we can't assign probabilities to points on real line. (It just doesn't work out mathematically.)
- Work with set of intervals, and Boolean operations on them.
- Set may be finite or countable.
- This set of events is a ''Borel set''.
- Notation:
- [a,b] closed. includes both. a<=x<=b
- (a,b) open. includes neither. a<x<b
- [a,b) includes a but not b, a<=x<b
- (a,b] includes b but not a, a<x<=b
- Assign probabilities to intervals (open or closed).
- E.g., uniform distribution on [0,1] \(P[a\le x\le b] = \frac{1}{b-a}\)
- Nonuniform distributions are common.
- 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.
- \(S = \{ x | 0\le x\le 1 \}\)
- \(p(x=1) = 1/2\)
- For \(0\le x_0 \le 1, p(x<x_0) = x_0/2\)
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For fun: Heads you win, tails... you win. You can beat the toss of a coin and here's how....
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Example 2.13, page 39, nonuniform distribution: chip lifetime.
- Propose that P[(t, \(\infty\) )] = \(e^{-at}\) for t>0.
- Does this satisfy the axioms?
- I: yes >0
- II: yes, P[S] = \(e^0\) = 1
- III here is more like a definition for the probability of a finite interval
- P[(r,s)] = P[(r, \(\infty\) )] - P[(s, \(\infty\) )] = \(e^{-ar} - e^{-as}\)
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Probability of a precise value occurring is 0, but it still can occur, since SOME value has to occur.
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Example 2.14: picking 2 numbers randomly in a unit square.
- Assume that the probability of a point falling in a particular region is proportional to the area of that region.
- E.g. P[x>1/2 and y<1/10] = 1/20
- P[x>y] = 1/2
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Recap:
- Problem statement defines a random experiment
- with an experimental procedure and set of measurements and observations
- that determine the possible outcomes and sample space
- Make an initial probability assignment
- based on experience or whatever
- that satisfies the axioms.