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SECTION 7.6PROBABILITY WITHCOMBINATION & PERMUTATIONS

Please review Ch 6.1, 6.2, and 6.4

REVIEW: PERMUTATIONS & COMBINATIONS

Permutations refers to the numbers of ways a group of objects canbe arranged, such that the order is important

Combinations refers to the numbers of ways a group of objectscan be arranged, such that the order is NOT important

“r” is the number of objects chosen &

“n – r” is the number of objects NOT chosen

Ex: with a deck of 52 cards, how many 5 card hands are possible?

Ex: A “flush” contains all 5 cards with the same suit. How manyflush hands are there?

Ex: How many hands with atleast 3 spades?

Need to consider the hands with 3, 4, and 5 spades

Ex: 5 people are in a race: Amy, Ben, Chris, Don, and Ed. Howmany different outcomes are possible?

Ex: If only 1st, 2nd, and 3rd places are taken, how many possibleoutcomes are there?

Ex: Given the letters: SWOOOSSHH, how many permutationsare possible?

PROBABILITY WITH PERMUTATIONS & COMBINATIONS:

When solving probability with counting principles involved:

first find the total number of desired outcomes and

Divide it by the total number of possible outcomes

Ex: What is the probability of being dealt the following five cardhands?

a) Flush with spades

b) Full house with 3 kings and 2 aces

Ex: Four cards is drawn from a deck, what is the probability ofnot getting any diamonds?

b) What is the probability of getting atleast one diamond?

Use the complement to find this probability:

Choose 4 cards from a deck of 52

39 cards that are not diamonds

Practice: What is the probability of getting a five card hand with3 hearts and 2 spades?

b) What is the probability of getting a seven card hand with 4 redcards and 3 spades?

c) What is the probability of getting 4 spades, 2 hearts, and 1 clubfrom a 7 card hand?

Use the bucket method to find the total number of desired hands

EX: THERE ARE 12 HORSES IN A RACE WITH 1ST, 2ND, AND 3RDPLACES. WHAT IS THE PROBABILITY THAT SOMEONE WILL GETALL THREE HORSES IN THE CORRECT ORDER?

Order matters, so its a permutation

Only 1 correct outcome

PRACTICE: IF THE ORDER OF THE HORSES IS NOT TAKEN INTOACCOUNT, WHAT IS THE PROBABILITY THAT SOMEONE WILLCORRECTLY GET ALL THREE HORSES?

Order doesn’t matters,so its a combination

Only 1 correct outcome

IN THE LOTTO 649 DRAW, 6 NUMBERS ARE DRAWN FROM 1TO 49 WITHOUT REPLACEMENT. WHAT IS THE PROBABILITYOF EACH EVENT?

a) Four out of six numbers are correct

b) Five out of six numbers are correct

Order doesn’t matters,so its a combination

43 other numbers that you didn’tchoose, 2 of them were in the lottery

Out of the 6 numbers you chose,4 of them are correct

Ex: When given a five card hand, what is the probability ofgetting a full house ( 3 of a kind and a pair)

Note: there are 13 different values to choose for the pair and 3 of a kind

Out of the 13 values, there are 13 choices for the triple

For each value, there are 4 suits: (Spade, hearts, diamonds and clubs)

For the three of a kinds, out of the 4 suits, choose 3

For the pair, out of the 4 suits, choose 3

Then there are 12 values left for the pair

Challenge: When dealt a five card hand, what is theprobability of getting 2 pairs?

Note: there are 13 different values to choose for each pair and the single card

The first pair and second pair can be intertwined  13 choose 2

For each pair, there are four suits, and you choose two out of the four

For the single card, you have 4 cards and choose 1

For the single card, there are 11 options left, so multiply by 11

Alternative solution

Note: the difference between the fullhouse (permutation) and thetwo pairs(combination) is that the pairs can be in any order

HOMEWORK:

Pg. # 3, 5ab, 6-11, 13-16, 18, 20

PROBABILITY OF GETTING 2 PAIRS

Out of the 13 choices, choose 2 for the pair

Draw a bucket to separate each group

Out of the deck of 52 cards, 44 cards are left for thesingle card because it can not be the same as eitherone of the pairs

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