Counting Subsets of a Set: Combinations

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 numbers • There is only 1 winning combination. • Probability of winning is 1/13983816 = 0.00000007151.

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 0.00001845.

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) = 13545. • Probability is 0.0009686.

Lotto South • Match 0 of 6 numbers • Player 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.

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?

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.

Counting Subsets of a Set: Combinations