THEORY OF PROBABILITY
VLADIMIR KOBZAR
Theory of Probability: Lecture 5 - Equally Likely
Outcomes
Review Problems.
(1) For any event A, let an A-certificate be a contract that requires
the seller to give the buyer $1000 if the event A occurs, and
0 otherwise. For a particular (sports-based) experiment, your
friend is willing to buy or sell any A-certificate for Pf riend pAq ¨
1000 dollars, where Pf riend reflects your friend’s belief system.
Assume there are disjoint events B, C such that
Pf riend pBq ` Pf riend pCq ‰ Pf riend pB Y Cq.
(2)
(3)
(4)
(5)
Show how making a few trades with your friend can earn you
some money. This shows how a belief system that doesn’t adhere
to the axioms of probability can be exploited in the context of
betting.
You roll a fair n-sided die k-times.
(a) Give a sample space and an associated probability measure.
(b) What is the chance that your rolls are strictly increasing?
That is, the second roll is larger than the first, and third is
larger than the second, etc.
Model flipping two fair coins using a sample space and a probability measure. Compute the probability of getting at least 1
head.
Let S “ t1, 2, ..., 100u with all outcomes equally likely. What is
the probability of the event that a chosen integer isn’t divisible
by 2, 3, or 5?
We want to distribute 10 identical pieces of candy to 4 (distinguishable) children. For each piece of candy we roll a 4-sided
die and give it to the associated child. What is the probability
that the children receive 2, 5, 1, 2 pieces of candy, respectively
(i.e., first child receives 2, second receives 5, etc.)?
Date: July 16, 2016.
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VLADIMIR KOBZAR
(6) Suppose you are dealt an ordered sequence of k cards from a
52 card deck without replacement. Compute the probability of
getting a particular k card (unordered) hand.
Solutions.
(1) If Pf riend pBq ` Pf riend pCq ą Pf riend pB Y Cq, sell a B and Ccertificate, and purchase pB Y Cq certificate of the same notional.If Pf riend pBq ` Pf riend pCq ă Pf riend pB Y Cq, buy B and
C-certificates, and sell pB Y Cq certificate of the same notional.
In either case you, net a positive difference in the purchase price.
If B Y C occurs the cash flows offset each other. If B Y C does
not occur, there are no further payments. So, in all cases you
keep the net purchase price difference.
(2) (a) S “ tpd`1 ,˘...dk q : 1 ď di ď 6u where all outcomes are equally
likely. (b) nk {nk
(3) S “ tpa1 , a2 q : ai P tH, T uu where all outcomes are equally
likely. P pHT, T H, HHq “ 3{4
(4) Let D2 , D3 , D5 be the subsets of S divisible by 2,3,5. |D2 | “
50, |D3 | “ 33, |D5 | “ 20, |D2 D3 | “ 16, |D2 D5 | “ 10, |D3 D5 | “
6, |D2 D3 D5 | “ 3. By the exclusion inclusion principle, |D2 Y
D3 Y D5 | “ 50 ` 33 ` 20 ´ 16 ´ 10 ´ 6 ` 3 “ 74
(5) S “ tpd1 , ...d1 0q : 1 ď di ď 4u where all outcomes are equally
p 10 q
likely. 2,5,1,2
410
(6) Each ordered hand occurs with probability
1
52 ¨ 51 ¨ .... ¨ p52 ´ k ` 1q
Therefore each unordered hand occurs with probability
k!
1
“ `52˘
52 ¨ 51 ¨ .... ¨ p52 ´ k ` 1q
k
Select
by
Example
Type
mult
distribution
one
distribution
Model
Ordered
2.5b: described on the left.
6¨5¨4`5¨6¨4`5¨4¨6
4
“ 11
11¨10¨9
2.5d: described on the left. Ai
is the event that the special ball
is the ith ball chosen. P pAi q “
k
pn´1q...pn´i`1qp1qpn´iq...pn´
k `1q{pnpn´1q...pn´k `1qq “ n1
2.5f: P ptdealing a poker hand of 2.5j: P ptany given card appeardistinct consecutive values, not ing following the first ace when
5
51!
all in the same suituq “ 10p452´4q the whole deck is dealtuq “ 52!
p5q
2.5g: P ptdealing a poker hand
with three of a kind plus a
13¨12p42qp43q
pairuq “
p52
5q
p.33: P ptchoosing unrelated per- p.33:
described on the left
20¨18¨16¨14¨12
sons when choose 5 people from 20¨19¨18¨17¨16
p10q25
10 married couplesuq “ 520
p5q
2.5h(a): P ptdealing a bridge
hand with 13 spades to 1 of 4
4
playersuq “ 52
p13q
2.5h(b): P ptdealing an ace in
a bridge hand to each of 4
48
4!p
q
playersuq “ 12,12,12,12
52
p13,13,13,13q
Unordered
2.5b: P({drawing 1W & 2B balls
from a bowl
with 6W & 5B
p61qp52q
4
balls})= 11 “ 11
p3q
2.5d: P({drawing a special ball
by withdrawing k from n balls
p1qpn´1q
})= 1 nk´1 “ nk
2.5d: described on the left. Since
each of the balls is equally likely
to be chosen as the ith ball chosen P pAi q “ n1 . By mutual exclusivity, we get nk
Mutually Exclusive
THEORY OF PROBABILITY
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denominator is the number of off.-def. pairs times
the number of off. only
pairs and def. only pairs)
220 20!
Unordered
2.5k(i) : P ptpairing 20
off. and 20 def. players
as roommates s.t. there
are no off.-def. pairsuq “
20!
40!
2
p 210
10! q { 220 20!
2.5k(ii) : P t(pairing 20
off. and 20 def. players
as roommates s.t. there
are 2i off.-def. pairsuq “
2
p20´2iq!
2
p20
2i q p2iq!r 210´i p10´iq! s
(the
40!
2.5e: P ptordering of the
colors of successive balls
arranged from n red and
n!m!
m blueuq “ pn`mq!
Model
Ordered
2.5i:
P (no 2 of
n
people
celebrate
their birthday on the
same day of the year)
“ p365qp364q...p365´n`1q
365n
Notes: Example 2.5c is similar to 2.5b, and therefore not included.
Ordering
Example
Type
Matching
2.5o: P ptr runs of
wins for team with
n winsm`1
and m losses
p r qpn´1
r´1 q
uq “
pm`n
n q
Inclusion-Exclusion Integer Solutions
2.5m:
P ({none
of N men selects
his own hat}) “
řN
1
i
i“0 p´1q {i! Ñ e
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VLADIMIR KOBZAR