Probability 3.4
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Transcript of Probability 3.4
Counting PrinciplesCounting Principles Counting PrinciplesCounting Principles
Chapter 3.4Chapter 3.4
Objectives• Use the Fundamental Counting Principle
to find the number of ways 2 or more events can occur
• Find the number of ways a group of objects can be arranged in order
• Find the number of ways to choose several objects from a group without regard to order
• Use counting principles to find probabilities
The Fundamental Counting Principle
• If 1 event can occur in m ways and a second event can occur in n ways, the number of ways the 2 events can occur in sequence is m*n.
• This rule can be extended for any number of events occurring in sequence.
Buying a car• You can choose a Ford, Subaru, or
Porsche• You can choose a small or medium
car• You can choose red (R), purple (P),
white(W) or green (G)• How many choices do you have?• 3*2*3 = 18 choices
You can draw a tree diagram to show this
Security Systems• The access code for a car’s security
system consists of 4 digits. Each digit can be 0 through 9.
• How many access codes are possible if each digit can be used only once and not repeated?
• 10*9*8*7• = 5040
Security Systems• The access code for a car’s security
system consists of 4 digits. Each digit can be 0 through 9.
• How many access codes are possible if each digit can be repeated?
• 10*10*10*10• = 10,000
License Plates• How many license plates can you
make if a license plate consists of• Six (out of 26) letters each of
which can be repeated?• Six (out of 26) letters each of
which can not be repeated?
Permutations• A permutation is an ordered
arrangement of objects. • The number of different
permutations of n distinct objects is n!
• n! is read “n factorial”• 5! = 5*4*3*2*1• 0! is defined as 1
Baseball example• How many starting lineups are
possible for a 9 player baseball team?
• 9*8*7*6*5*4*3*2*1 • = 362,880• Where is the factorial key on • your calculator?
Finding nPr
• Find the number of ways of forming 3-digit codes in which no digit is repeated:
• Select the first digit – (10 choices)
• Select the second digit – (9 choices for each of the possible 10 first choices =
90)
• Select the third – (8 choices for each of the 90 previous choices = 720)
Permutations of n Objects taken r at a
Time• The number of permutations of n
distinct objects taken r at a time is
• nPr = n! , where r ≤ n
• (n-r)!• Read this “Probability of n choose
r”
Permutations of n Objects taken r at a
Time• Use the formula for the last
problem
• nPr = n! , where r ≤ n
• (n-r)!
• 10P3 = 10! =10*9*8*7*6*5*4*3*2*1
• (10-3)! 7*6*5*4*3*2*1
Permutations of n Objects taken r at a
Time
Another example . . .
AAAB• How many ways can this be
arranged?• AAAB• AABA• ABAA• BAAA• 4
AAAABBC• How many ways can this be
arranged?• 105, but it would be tedious
writing these all out, so let’s use a formula:
Distinguishable Permutations
Try it with AAAB:• 4! = 4*3*2*1 = 4• 3!*1! 3*2*1
Try it with AAAABBC• 7! = 7*6*5 = 105• 4!*2!*1! 2
Combinations• You want to buy 3 hats from a selection
of 5. How many possible choices do you have?
• ABC ABD ABE• ACD ACE• ADE• BCD BCE
• BDE CDE
10 Choices
Combinations• A combination is a selection of r objects
from a group of n objects without regard to order and is denoted by nCr.
• The number of combinations of r objects selected from a group of n objects is
• nCr = = n! .
• (n-r)!r!
Combinations• Try this with the 5 hats, choose 3:
• nCr = = n! .
• (n-r)!r!
• nCr = 5! . = 5*4*3*2*1 = 10
• (5-3)!*3! 2*1*3*2*1
MISSISSIPPI• A word consists of 1 M, 4 I’s, 4 S’s, and 2 P’s.
If the letters are randomly arranged in order, what is the probability that the arrangement spells out Mississippi?
• How many distinguishable permutations?• 11! =• 1!*4!*3!*2!• 34,650• Probability(Mississippi) = 1/34,650 = .000029
Diamond Flush• Find the probability of being dealt 5 diamonds
from a standard deck of cards.• How many ways to choose 5 out of 13 diamonds?
• 13C5
• How many ways to choose a 5 card hand?
• 52C5
• P(diamond flush) = 13C5 = 1,287 = .0005
• 52C5 = 2,598,960
Homework• P. 157 12-40 every 4th problem
Now try this . . .• With Skittles!