ECSE-2500, Engineering Probability, Spring 2010, Rensselaer Polytechnic Institute

Lecture 4

Example 2.13, page 39, nonuniform distribution: chip lifetime.
1. Propose that P[(t,infinity)] = exp(-at) for t>0.
2. Does this satisfy the axioms?
3. I: yes >0
4. II: yes, P[S] = exp(0) = 1
5. III here is more like a definition for the prob of a finite interval
6. P[(r,s)] = P[(r,infinity)] - P[(s,infinity)] = exp(-ar) - exp(-as)
Prob of a precise value occurring is 0, but it still can occur, since SOME value has to occur.
Example 2.14: picking 2 numbers randomly in a unit square.
1. Assume that the prob 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
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.
2.3 Counting methods
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 $}
2.3.1 Sampling WITH replacement and WITH ordering
1. Consider an urn with n different colored balls.
2. Repeat k times:
  1. Draw a ball.
  2. Write down its color.
  3. Put it back.
3. # Distinct ordered k-tuples = {$n^k$}
Example 2.1.5. How many distinct ordered pairs for 2 balls from 5? 5*5.
iClicker. 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?
- A 16
- B 12
- C 8
- D 4
- something else
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. # Distinct ordered k-tuples = n(n-1)(n-2)...(n-k+1)
iClicker. 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?
- A 16
- B 12
- C 8
- D 4
- something else
Example 2.1.6: Draw 2 balls from 5 w/o replacement.
1. 5 choices for 1st ball, 4 for 2nd. 20 outcomes.
2. Prob that 1st ball is larger?
3. List the 20 outcomes. 10 have 1st ball larger. P=1/2.
Example 2.1.7: Draw 3 balls from 5 with replacement. What's the probability they're all different?
1. P = (# cases where they're different)/(# cases where I don't care)
2. P = (# case w/o replacement)/(# cases w replacement)
3. P = (5*4*3)/(5*5*5)
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: {$$ n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n\left(1+\frac{1}{12n}+...\right) $$}
6. Therefore if you ignore the last term, the relative error is about 1/(12n).
Example 2.1.8. # permutations of 3 objects. 6!
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. # events for all 12 crashes = {$ 12^{12} $}
3. # events for 12 crashes in 12 different months = 12!
4. Prob = {$ 12!/(12^{12}) = 0.000054 $}
5. Random does not mean uniform.
2.3.4 Sampling w/o replacement and w/o ordering
1. We care what objects we pick but not the order
2. E.g., drawing a hand of cards.
3. term: Combinations of k objects selected from n. Binomial coefficient. {$ C^n_k = {n \choose k} = \frac{n!}{k! (n-k)!} $}
4. Permutations is when order matters.
Example 2.20. Select 2 from 5 w/o order. {$ 5\choose 2 $}
Example 2.21 # permutations of k black and n-k white balls. This is choosing k from n.
Example 2.22. 10 of 50 items are bad. What's probability 5 of 10 selected randomly are bad?
1. # ways to have 10 bad items in 50 is {$ 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.
Multinomial coefficient: Partition n items into sets of size {k_1, k_2, ... k_j, \sum k_i=n $} {$ \frac{n!}{k_1! k_2! ... k_j!} $}
2.3.5. skip
Conditional probability
1. big topic
2. E.g., if it snows today, is it more likely to snow tomorrow? next week? in 6 months?
3. E.g., what is the probability of the stock market rising tomorrow given that (it went up today, the deficit went down, an oil pipeline was blown up, ...)?
4. definition {$ P[A|B] = \frac{P[A\cap B]}{P[B]} $}
Example 2.2.4 p 48. 4 balls {1b,2b,3w,4w}
1. event A: black
2. B: even
3. C: >2
4. P[A{$ \cap $}B], P[A{$ \cap $}C], P[A|B], P[A|C]
P[A{$\cap$}B] = P[A|B]P[B] = P[B|A]P[A]
Example 2.25 2 black, 3 white balls. Draw 2 w/o replacement. What's prob both are black? Use a tree diagram.
Example 2.26 Binary communication. Source xmits 0 w.p. (1-p) and 1 w.p. p. Receiver errs w.p. e. What are probabilities of 4 events?
Total probability theorem
1. {$B_i$} mutually exclusive events whose union is S
2. P[A] = P[A {$\cap B_1$} + P[A {$\cap B_2$} + ...
3. {$ P[A] = P[A|B_1]P[B_1] + P[A|B_2]P[B_2] + ... $}
Example 2.28. Chip quality control. P[good]=(1-p), P[bad]=p.
1. Good: P[still alive at t] = exp(-at); bad: exp(-1000at).
2. What's the prob that a random chip is still alive at t?