ProbabilityProbability Counting Outcomes and Theoretical Probability

ProbabilityProbabilityProbabilityProbability

Counting Outcomes and Counting Outcomes and Theoretical ProbabilityTheoretical Probability

What is Probability?• the relative frequency with which

an event occurs or is likely to occur

• typically expressed as a ratio

Counting Principle• to determine the total number of

possibilities that can occur in an event

Example #1• A store sells caps in three colors

(red, white, and blue), two sizes (child and adult), and two fabrics (wool and polyester).

• How many cap choices are there?• (colors)(sizes)(fabrics)• (3)(2)(2) = 12 choices

Example #2• How many three letter monograms

are possible in the English language?

• (1st letter) (2nd letter) (3rd letter)• (26)(26)(26)• 17576

Theoretical Probability• number of favorable outcomes number of possible outcomes

why is it called theoretical?

Example #3• What is the probability of winning

the Play-4 lottery if you purchase two tickets with different numbers?

(1st digit)(2nd digit)(3rd digit)(4th digit)(10)(10)(10)(10)100002:10,000 or 1:5,000

Independent Events• Events that do not have an affect

on one another– Tossing a coin multiple times– Rolling a die multiple times– Repeating digits or letters– Replacing between events

Dependent Events• One event happening affects

another event happening– Not replacing between events– No repeating digits or letters

P(A, then B)• The probability of event A

occurring then the probability of event B occurring

• Total probability is determined by multiplying the individual probabilities

Example #1• You choose a card from a regular

deck of playing cards. After returning it to the deck, you choose a second card. What is the probability that the first card will be red and the second card will be a seven?

• 52 cards in the deck• 26 red and 26 black• 4 sevens

1

26

26

52

4

52 1

13

1

2

Example #2• You choose a card from a regular

deck of playing cards. Without returning it to the deck, you choose a second card. What is the probability that the first card will be a red face card and the second card will be a seven?

• 52 cards in the deck• 26 red and 26 black• 12 face cards• 4 sevens

2

221

6

52

4

51 4

51

3

26

Example #3• There are five girls and two boys

seated in a waiting room. What is the probability that the first person called will be a girl and the second one called will be a boy?

• 5 girls• 2 boys

5

21

5

7

2

6 1

3

5

7

Example #4• There are five girls and two boys

seated in a waiting room. What is the probability that the first person called will be a boy and the second one called will be a girl?

• 5 girls• 2 boys

5

21

2

7

5

6

Permutations• determining the number of

arrangements of items when the order is important

• nPr– P → permutations– n → number of objects– r → number chosen

Example #1• 8 people wish to buy tickets for a

concert. In how many ways could the first five members get in line?

• 8P5 = (8)(7)(6)(5)(4)

• 8P5 = 6720

Example #2• How many arrangements of four

books can be made from a stack of nine books on a shelf?

• 9P4 = (9)(8)(7)(6)

• 9P4 = 3024

Combinations• determining the number of

arrangements of items when the order is not important

• nCr– C → combinations– n → number of objects– r → number chosen

Example #3• You have five choices of sandwich

fillings. How many different sandwiches can you make using three of the fillings?

• 5C3 = 5P3/3P3

• 5C3 = (5)(4)(3)/(3)(2)(1)

• 5C3 = 10