Transcript Probability

Probability
Counting Methods
Do Now:
• Find the probablity of:
• Rolling a 4 on a die
• Rolling a “hard eight” (2 - 4’s) on a pair of
dice.
Counting principal…
• Find the probability of rolling “snake eyes”
on a pair of dice
1 1 1
 
6 6 36
Vocabulary:
ordering
A permutation is an ___________________________
of
objects. The number of permutations of r objects taken
from a group of n distinct objects is denoted nPr. (order
matters)
selection
• A combination is a _________________________
of r
objects from a group of n objects where the order is not
important. The number of r objects taken from a group
of n distinct objects is denoted nCr.
Fundamental Counting Principle
• If one event can occur in m ways and another event can
occur in n ways then the total number of ways both
m*n
events can occur is _____________.
• Three or More Events – the fundamental counting
principle can be extended to three or more events. For
example, if three events can occur in m, n, and p ways,
then the number of ways that all three events can occur
m*n*p
is __________________.
Example 1
• You are buying a sandwich. You have a choice of 5
meats, 4 cheeses, 3 dressings, and 8 other toppings.
How many different sandwiches with one meat, one
cheese, one dressing, and one other topping can you
choose?
Example 2
• A town has telephone numbers that begin with 432 and
437 followed by four digits. How many different
telephone numbers are possible if the last four digits
cannot be repeated?
Example 2
• A town has telephone numbers that begin with 432 and
437 followed by four digits. How many different
telephone numbers are possible if the last four digits
cannot be repeated?
Example 3
• Twenty-six golfers are competing in the final round of a
local competition. How many different ways can 3 of the
golfers finish first, second, and third?
Example 3
• Twenty-six golfers are competing in the final round of a
local competition. How many different ways can 3 of the
golfers finish first, second, and third?
Permutations
• Permutations Of n Objects Taken r at a Time
The number of permutations when order is important:
P
n r
• Permutations with Repetition
The number of permutations where one object is
repeated q1 times, another is repeated q2 times and so
on (like repeated letters in a word arrangement)
n!
q1 !  q2 !  ... qk !
Example 4
• Find the value of
7P3.
Example 5:
• Using the digits 2,3,4,5,6 how many 3-digit
numbers can be formed if repetition of digits is
not permitted?
Example 6:
Find the number of distinguishable permutations of the
letters in
(a) ALGEBRA
(b) MATHEMATICS.
Combinations Of n Objects Taken r at a Time
The number of combinations of where order is not important.
n
Cr
(This number is always smaller than the corresponding number of
permutations) This is due to each combination yielding a number of
permutations (two for each combination, for example, AB and BA).
Example 7: Find the value of 10 C 2 .
Example 8
• For a history report, you can choose to write about 3 of
the original 13 colonies. How many different
combinations exist for the colonies you will be writing
about?
Example 9
– The standard configuration for NJ license plates today
is 3 letters, 2 digits, and 1 letter. How many different
license plates are possible if letters and digits can be
repeated?
– How many different license plates are possible if
letters and digits cannot be repeated?
Start with:
– The standard configuration for NJ license plates today
is 3 letters, 2 digits, and 1 letter. How many different
license plates are possible if letters and digits can be
repeated?
– ____ ____ ____ ____ ____ ____
– How many different license plates are possible if
letters and digits cannot be repeated?
– ____ ____ ____ ____ ____ ____
End with:
– The standard configuration for NJ license plates today
is 3 letters, 2 digits, and 1 letter. How many different
license plates are possible if letters and digits can be
repeated?
26  26  26 10 10  26  45,697,600
– How many different license plates are possible if
letters and digits cannot be repeated?
26  25  24 10  9  23  32,292,000
Probability binomial formula
• Memorize:
r
C
p
q
n r
•
•
•
•
nr
p = probability of success
q = probability of failure
n = number of trials
r = number of successes
Example:
• Brianna makes about 90% of the shots on
goal she attempts. Find the probability that
Bri makes exactly 7 out of 12 goals.
• Since you want 7 successes (and 5
failures), use the term p7q5.
• This term has the coefficient 12C7.
Apply the formula:
• Probability (7 out of 12) = 12C7 p7q5
• =
792 • (0.9)7(0.1)5
(p = 90%, or 0.9)
• = 0.0037881114
• Bri has about a 0.4% chance of making
exactly 7 out of 12 consecutive goals.
Example 2:
• A fair die is tossed five times. What is the
probability of tossing a 6 exactly 3 times.
• Name p,q,n,r
Example 2:
• A fair die is tossed five times. What is the
probability of tossing a 6 exactly 3 times.
1
p
6
5
q
6
n5
r3
Apply this to the formula:
r
C
p
q
n r
nr
Example 2:
• A fair die is tossed five times. What is the
probability of tossing a 6 exactly 3 times.
1
6
5
q
6
n5
r3
p
1
5 C3  
6
3
 3%
5
1 25

 .032150206
   10 
6
216 36
2
At least/at most
• At least- that number or more
• At most- that number or less (including 0)
• Examples:
• at most 3 out of 5 means 3, 2, 1, or 0
• At least 3 out of 5 means 3, 4, or 5
An exactly example:
1
• The probability of Chris getting a hit is
3
If he comes to bat four times, what is the
probability that he gets exactly 2 hits?
r
n
Cr p q
nr
An exactly example:
1
• The probability of Chris getting a hit is
3
If he comes to bat four times, what is the
probability that he gets exactly 2 hits?
1 2
1 4 24 8
 

4 C2      6 
 3  3 
9 9 81 27
2
2
At least example:
1
• The probability of Chris getting a hit is
3
If he comes to bat four times, what is the
probability that he gets at least 3 hits?
Find:
Prob (3 hits) + prob (4 hits)
At least example:
1
• The probability of Chris getting a hit is
3
If he comes to bat four times, what is the
probability that he gets at least 3 hits?
Find:
Prob (3 hits) + prob (4 hits)
1
4 C3  
 3
3
2
1
  4 C4  
 3
 3
1
4
2
 
 3
0
At least example:
1
• The probability of Chris getting a hit is
3
If he comes to bat four times, what is the
probability that he gets at least 3 hits?
Find:
Prob (3 hits) + prob (4 hits)
1
4 C3  
 3
3
2
1
   4 C4  
 3
 3
1
4
2
8 1 1
    
 3  81 81 9
0
At most example:
1
• The probability of Chris getting a hit is
3
If he comes to bat four times, what is the
probability that he gets at most 1 hit?
Find:
Prob (1 hit) + prob (0 hits)
At most example:
1
• The probability of Chris getting a hit is
3
If he comes to bat four times, what is the
probability that he gets at most 1 hit?
Find:
Prob (1 hit) + prob (0 hits)
1
4 C1  
 3
1
2
1
   4 C0  
 3
 3
3
0
 2  32 16 48 16
 

  
 3
81 81 81 27
4