Template for MCMA Poster Slides

1
Correction Problem 18

Alice and Bob have 2n1 FAIR coins, each with
probability of a head equal to 1/2
2
Review

• Conditional probability of A given B , P(B)gt0
• P(AB)P(AnB)/P(B)
• Define a new probability law
• Conditional independence
• P(AnCB)P(AB) P(CB)
• Total probability theorem
• Bayes rule

3
1.6 Counting

4
Counting

• To calculate the number of outcomes
• Examples
• To toss a fair coin 10 times, whats the
  probability that the first toss was a head?
• Fair coin 1/2
• To toss a fair coin 10 times, whats the
  probability that there was only 1 head?
• 1/10?
• What about the probability that there are 5
  heads?
• 1/2?

5
The Counting Principle

• An r stage process
• (a) n1 possible results at the first stage
• (b) For every possible result of the first stage,
  there are n2 possible results at the second stage
• (c) ni ..
• Total number of possible results n1 n2 nr
  (proof by induction)

6
The Counting Principle

• Example1 fair coin toss 3 times
• 2X2X2 8 possible outcomes
• TTT TTH THT THH HTT HTH HHT HHH
• Example2 number of subsets of an n-element set
  S
• S1,2
• Subsets ?, 1, 2, 1,2, 422 subsets
• S1,2,3
• Subsets ?, 1, 2, 1,2, 3, 1,3,
  2,3, 1,2,3, 823 subsets
• The choice of a subset as a sequential process of
  choosing one element at a time.
• n stages, binary choice at each stage
• 2X2X2X2 2n

7
Permutations

• Selection of k objects out of n objects (kltn),
  order matters
• Sequences 123? 321
• K-permutation the number of possible sequence
• Example number of 3-letter words using a,b,c or
  d at most once
• 4X3X2 24 3-permutation out of 4 objects

8
Permutations (continued)

• Selection of k objects out of n objects (kltn),
  order matters
• k-stages, n1n, ni1 ni-1, nk n-k1
• Counting principle n(n-1)(n-2)(n-k1)
• number of permutations of n objects out of n
  objects (kn)
• n! (0! 1)

9
Combinations

• Selection of k objects out of n objects (kltn),
  NO ordering
• Sets 1,2,3 3,2,1
• k-combinations the number of possible different
  K-element subsets
• Example number of 3-element subsets of
  a,b,c,d.
• a,b,c a,b,d a,c,d b,c,d 4 3-element
  subsets
• 3-permutations of a,b,c,d abc, acb, bac, bca,
  cab, cba,
  abd, adb, bad,bda, dab, dba,
  acd, adc, cad, cda,
  dac, dca bcd,
  bdc, cbd, cdb, dbc, dcb
• k-combinations k-permutations Order
• Each set (k-combination) is counted k! times in
  the k-permutation.

10
Combinations (continued)

• Example1 2-combinations of a 4-object set
  a,b,c,d.
• 4 choose 2 4!/(2!2!) 6
• a,b a,c a,d b,c b,d, c,d 6
  2-element subsets
• Example2 k-head sequences of n coin tosses
• n5, k2
• HHTTT HTHTT HTTHT HTTTH THHTT
• THTHT THTTH TTHHT TTHTH TTTHH
• HHTTT ? 1,2 HTHTT ? 1,3. TTTHH ?4,5
• Number of k-head sequences number of
  k-combinations from 1,2,n

11
Combinations (continued)

• Properties of n_choose_k

12
Expansion of (ab)n

13
Combinations (continued)

• Binomial formula
• Let p 1/2

14
Partitions

• Partitions of n objects into r groups, with the
  ith group having ni objects, sum of ni is equal
  to n.
• Order does not matter within a group, the groups
  are labeled
• Example partition S1,2,3,4,5 into 3 groups
  n1 2, n2 2, n3 1
• 1,23,45 2,14,35
• 1,23,45 ? 3,41,25
• Total number of choices (group by group)

15
Partitions (continued)

• Example partitions of 4 objects into 3 groups,
  2-1-1.
• 4!/(2!1!1!)12
• abcd abdc
• acbd acdb
• adbc adcb
• bcad bcda
• bdac bdca
• cdab cdba

16
Summary

• k-permutation of n objects n!/(n-k)!
• Order matters 123 ? 321
• k-combinations of n objects
• Order does not matter 1,2,3 3,2,1
• Partitions of n objects into r groups, with the
  ith group having ni objects
• Order does not matter within a group, the groups
  are labeled

17
Binomial formula

• Toss an unfair coin (p-head, (1-p)-tail) n times
• The outcome is a n-sequence THHTHTH
• OHHH,HHT,HTH,HTT,THH,THT,TTH,TTT for n3
• Group the sequences according to the number of H

18
Pascals Triangle