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

# Lecture 4

- Example 2.13, page 39, nonuniform distribution: chip lifetime.
- Propose that P[(t,infinity)] = exp(-at) for t>0.
- Does this satisfy the axioms?
- I: yes >0
- II: yes, P[S] = exp(0) = 1
- III here is more like a definition for the prob of a finite interval
- 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.
- Assume that the prob 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

- 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.

- 2.3 Counting methods
- finite sample space
- each outcome equally probable
- get some useful formulae
- warmup: consider a multiple choice exam where 1st answer has 3 choices,
2nd answer has 5 choices and 3rd answer has 6 choices.
- Q: How many ways can a student answer the exam?
- A: 3x5x6

- 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
- Consider an urn with n different colored balls.
- Repeat k times:
- Draw a ball.
- Write down its color.
- Put it back.

- # 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
- Consider an urn with n different colored balls.
- Repeat k times:
- Draw a ball.
- Write down its color.
- Don't put it back.

- # 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.
- 5 choices for 1st ball, 4 for 2nd. 20 outcomes.
- Prob that 1st ball is larger?
- 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?
- P = (# cases where they're different)/(# cases where I don't care)
- P = (# case w/o replacement)/(# cases w replacement)
- P = (5*4*3)/(5*5*5)

- 2.3.3 Permutations of n distinct objects
*Distinct*means that you can tell the objects apart.- This is sampling w/o replacement for k=n
- 1.2.3.4...n = n!
- It grows fast. 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 7!=5040
- Stirling approx: {$$ n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n\left(1+\frac{1}{12n}+...\right) $$}
- 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?
- For each crash, let the outcome be its month.
- # events for all 12 crashes = {$ 12^{12} $}
- # events for 12 crashes in 12 different months = 12!
- Prob = {$ 12!/(12^{12}) = 0.000054 $}
**Random does not mean uniform.**

- 2.3.4 Sampling w/o replacement and w/o ordering
- We care what objects we pick but not the order
- E.g., drawing a hand of cards.
- term:
**Combinations**of k objects selected from n. Binomial coefficient. {$ C^n_k = {n \choose k} = \frac{n!}{k! (n-k)!} $} **Permutations**is when order matters.

- Example 2.20. Select 2 from 5 w/o order. {$ 5\choose 2 $}
- Example 2.21 # permutations of k black and n-k white balls. This is choosing k from n.
- Example 2.22. 10 of 50 items are bad. What's probability 5 of 10
selected randomly are bad?
- # ways to have 10 bad items in 50 is {$ 50\choose 10$}
- # ways to have exactly 5 bad is 3 ways to select 5 good from 40 times # ways to select 5 bad from 10 = {$ {40\choose5} {10\choose5} $}
- 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!} $}
- 2.3.5. skip
- Conditional probability
- big topic
- E.g., if it snows today, is it more likely to snow tomorrow? next week? in 6 months?
- E.g., what is the probability of the stock market rising tomorrow given that (it went up today, the deficit went down, an oil pipeline was blown up, ...)?
- definition {$ P[A|B] = \frac{P[A\cap B]}{P[B]} $}

- Example 2.2.4 p 48. 4 balls {1b,2b,3w,4w}
- event A: black
- B: even
- C: >2
- 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
- {$B_i$} mutually exclusive events whose union is S
- P[A] = P[A {$\cap B_1$} + P[A {$\cap B_2$} + ...
- {$ 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.
- Good: P[still alive at t] = exp(-at); bad: exp(-1000at).
- What's the prob that a random chip is still alive at t?