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 web page

Lotto South There are C(49, 6) = 13,983,816 possible choices. Match all 6 numbers There is only 1 winning combination. Probability of winning is 1/ =

Lotto South Match 5 of 6 numbers There 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

Lotto South Match 4 of 6 numbers Player chooses 4 winning numbers and 2 losing numbers. Number of ways is C(6, 4)  C(43, 2) = Probability is

Lotto South Match 3 of 6 numbers Player chooses 3 winning numbers and 3 losing numbers. Number of ways is C(6, 3)  C(43, 3) = Probability is

Lotto South Match 2 of 6 numbers Player chooses 2 winning numbers and 4 losing numbers. Number of ways is C(6, 2)  C(43, 4) = Probability is

Lotto South Match 1 of 6 numbers Player chooses 1 winning numbers and 5 losing numbers. Number of ways is C(6, 1)  C(43, 5) = Probability is

Lotto South Match 0 of 6 numbers Player chooses 6 losing numbers. Number of ways is C(43, 6) = Probability is

Lotto South Note also that the sum of these integers is 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) = / =

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.

Another Lottery In the previous lottery, the probability of winning a cash prize is 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  = $1000  = $75  = $5  = $0  =

Lotto South The total is $0.8945, or 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 n 1 indistinguishable elements of one type, n 2 indistinguishable elements of another type, and so on, through k types, where n 1 + n 2 + … + n k = n. Then the number of (distinguishable) permutations of the n elements is n!/(n 1 !n 2 !…n k !).

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, n 1 ) choices of where to place the elements of the first type.

Proof of Theorem Proof: Then there are C(n – n 1, n 2 ) choices of where to place the elements of the second type. Then there are C(n – n 1 – n 2, n 3 ) 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, n 1 )  C(n – n 1, n 2 )  C(n – n 1 – n 2, n 3 ) … C(n – n 1 – n 2 – … – n k – 1, n k ) = …(some algebra)… = n!/(n 1 !n 2 !…n k !).

Example How many different numbers can be formed by permuting the digits of the number ?

Example How many permutations are there of the letters in the word MISSISSIPPI? How many for VIRGINIA? How many for INDIVISIBILITY?

Poker Hands Two of a kind. Two pairs. Three of a kind. Straight. Flush. Full house. Four of a kind. Straight flush. Royal flush.