lecture 13

Question 1

DESCRIBE symmetry argument ex - coins

Answer 1

single coin flip
usually can assume that 1/2 for heads and tails

Question 1

describe numerical probability - for equally likely sample outcomes

Answer 1

S = finite, N sample outcomes, equally likely
for elementary events E1,E2,…,EN we have P (Ei)= 1/N I = 1,…,n
FOR A some or all in S =
P (A) = number of sample outcomes in A/number of sample outcomes in S = n/N

Question 2

DESCRIBE symmetry argument ex - dice

Answer 2

for one event = P(Ei) = 1/6
for even = E2 union E4 union E6 so P(A) = 3/6 = 1/2

Question 3

when not equally likely what do we use for probably

Answer 3

relative frequency

Question 4

descrive relative frequency

Answer 4

define P(A) by considering the relative frequency
with which event A occurs in a long sequence of repeated
identical experiments
N repeated experiments
n is number of times/N that A occurs (like calc limit)
FREQUENTIST def of probability

Question 5

describe frequentist def - thumbtack ex

Answer 5

keep tossing and eventually will learn probabilities

Question 6

Is it always possible to consider an infinite sequence of repeats

Answer 6

no
ex = millionth digit of pi
can only be a number from 0-9

Question 7

rules for counting operations

Answer 7

A sequence of k operations, in which operation i can result in ni possible outcomes can result in n1 x n2 x … nk
possible sequence of outcomes
ex = any number from 0-9

Question 8

give ex of counting operations - gen

Answer 8

2 dice rolled = 6x6 = 36 possible outcomes
k=5, any numbers between 1-100 = 100^5 possible outcomes
k= 11, players of sport, 25x24x23… = PLAYERS cannot be picked twice

Question 9

describe selecting with replacement

Answer 9

each successive selection can be one of original set, regardless of previous selections

Question 10

describe selecting without replacement

Answer 10

set is depleted by each successive selection

Question 11

describe permutations

Answer 11

ordered arrangement of r objects
number of ways of ordering r objects selected, without replacement =
n
P
r
= n!/(n-r)!
(factorials)

Question 12

describe combinations

Answer 12

number of combinations of n objects taken r at a time is the number of subsets of size r that can be formed
n
C
r
= (n
r)
where
(n
r)
= n!/r!(n-r)!
i.e. we choose r from n

Question 13

describe number of possible selections of r objects

Answer 13

in sequence without replacement
n
P
r
= n!/(n-r)!

leaving (n-r) objects unselected

Question 14

describe number of possible selections of r objects - if order of selected objects not important in identifying combo we must have that

Answer 14

pnr = r! x cnr
there are r! equivalent combos that yield same permutation
r! turns unordered selection into ordered

Question 15

describe binary sequences

Answer 15

if take 1 = inclusion and 0 = exclusion then we can identify combos with binary sequence
the number of binary sequences of length n containing r 1s is
n choose r
(n
r)

Question 16

describe how to compute probabilities in case of equally likely outcomes

Answer 16

S: complete list of possible sequences of selections.
A: sequences having property of interest.
P (A)= nA/nS
use counting rules to count A & S