counting subsets of a set: combinations lecture 33 section 6.4 tue, mar 27, 2007
TRANSCRIPT
Counting Subsets of a Set: Combinations
Lecture 33
Section 6.4
Tue, Mar 27, 2007
Lotto South
In Lotto South, a player chooses 6 numbers from 1 to 49.
Then the state chooses at random 6 numbers from 1 to 49.
The player wins according to how many of his numbers match the ones the state chooses.
See the Lotto South web page.
Lotto South
There are C(49, 6) = 13,983,816 possible choices.
Match all 6 numbersThere is only 1 winning combination.Probability of winning is
1/13983816 = 0.00000007151.
Lotto South
Match 5 of 6 numbersThere are 6 winning numbers and 43 losing
numbers.Player chooses 5 winning numbers and 1
losing numbers.Number of ways is C(6, 5) C(43, 1) = 258.Probability is 0.00001845.
Lotto South
Match 4 of 6 numbersPlayer chooses 4 winning numbers and 2
losing numbers.Number of ways is C(6, 4) C(43, 2) =
13545.Probability is 0.0009686.
Lotto South
Match 3 of 6 numbersPlayer chooses 3 winning numbers and 3
losing numbers.Number of ways is C(6, 3) C(43, 3) =
246820.Probability is 0.01765.
Lotto South
Match 2 of 6 numbersPlayer chooses 2 winning numbers and 4
losing numbers.Number of ways is C(6, 2) C(43, 4) =
1851150.Probability is 0.1324.
Lotto South
Match 1 of 6 numbersPlayer chooses 1 winning numbers and 5
losing numbers.Number of ways is C(6, 1) C(43, 5) =
3011652.Probability is 0.4130.
Lotto South
Match 0 of 6 numbersPlayer chooses 6 losing numbers.Number of ways is C(43, 6) = 2760681.Probability is 0.4360.
Lotto South
Note also that the sum of these integers is 13983816.
Note also that the lottery pays out a prize only if the player matches 3 or more numbers.Match 3 – win $5.Match 4 – win $75.Match 5 – win $1000.Match 6 – win millions.
Lotto South
Given that a lottery player wins a prize, what is the probability that he won the $5 prize?
P(he won $5, given that he won)
= P(match 3)/P(match 3, 4, 5, or 6)
= 0.01765/0.01864
= 0.9469.
Example
Theorem (The Vandermonde convolution): For all integers n 0 and for all integers r with 0 r n,
Proof: See p. 362, Sec. 6.6, Ex. 18.
r
k r
n
kr
rn
k
r
0
Another Lottery
In the previous lottery, the probability of winning a cash prize is 0.018637545.
Suppose that the prize for matching 2 numbers is… another lottery ticket!
Then what is the probability of winning a cash prize?
Lotto South
What is the average prize value of a ticket? Multiply each prize value by its probability
and then add up the products:$10,000,000 0.00000007151 = 0.7151$1000 0.00001845 = 0.0185$75 0.0009686 = 0.0726$5 0.01765 = 0.0883$0 0.9814 = 0.0000
Lotto South
The total is $0.8945, or 89.45 cents (assuming that the big prize is ten million dollars).
A ticket costs $1.00. How large must the grand prize be to make
the average value of a ticket more than $1.00?
Another Lottery
What is the average prize value if matching 2 numbers wins another lottery ticket?
Permutations of Sets with Repeated Elements
Theorem: Suppose a set contains n1 indistinguishable elements of one type, n2 indistinguishable elements of another type, and so on, through k types, where
n1 + n2 + … + nk = n.
Then the number of (distinguishable) permutations of the n elements is
n!/(n1!n2!…nk!).
Proof of Theorem
Proof: Rather than consider permutations per se,
consider the choices of where to put the different types of element.
There are C(n, n1) choices of where to place the elements of the first type.
Proof of Theorem
Proof: Then there are C(n – n1, n2) choices of
where to place the elements of the second type.
Then there are C(n – n1 – n2, n3) choices of where to place the elements of the third type.
And so on.
Proof, continued
Therefore, the total number of choices, and hence permutations, is
C(n, n1) C(n – n1, n2) C(n – n1 – n2, n3) … C(n – n1 – n2 – … – nk – 1, nk)
= …(some algebra)…
= n!/(n1!n2!…nk!).
Example
How many different numbers can be formed by permuting the digits of the number 444556?
60!1!2!3
!6
Example
How many permutations are there of the letters in the word MISSISSIPPI?
How many for VIRGINIA? How many for INDIVISIBILITY?
34650!1!2!4!4
!11
Poker Hands
Two of a kind. Two pairs. Three of a kind. Straight. Flush. Full house. Four of a kind. Straight flush. Royal flush.