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

6•16

=136

Vocabulary:Vocabulary:

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)

• 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.

ordering

selection

Fundamental Counting PrincipleFundamental 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 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 is __________________.

m * n

m * n * p

Example 1Example 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 2Example 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 2Example 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 3Example 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 3Example 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?

PermutationsPermutations• Permutations Of n Objects Taken r at a Time

The number of permutations when order is important:

• 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 Pr

1 2 k

n

q q q

!

! ! ... !

Example 4Example 4

• Find the value of 7P3.

Example 5: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: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.

(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).

Combinations Of n Objects Taken r at a Time

The number of combinations of where order is not important.

(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 2C

n Cr

Example 8Example 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 9Example 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: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: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?

– How many different license plates are possible if letters and digits cannot be repeated?

26 • 26 • 26 • 10 • 10 • 26 =45,697,600

26 • 25 • 24 • 10 • 9 • 23=32,292,000

Probability binomial formula

• Memorize:

• p = probability of success

• q = probability of failure

• n = number of trials

• r = number of successes

n Cr

prqn−r

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.

p =16

q=56

n=5r=3

Apply this to the formula:

n Cr

prqn−r

Example 2:

• A fair die is tossed five times. What is the probability of tossing a 6 exactly 3 times.

p =16

q=56

n=5r=3

5C3

1

6

⎛

⎝⎜⎞

⎠⎟3 5

6⎛

⎝⎜⎞

⎠⎟2

=10 •1

216•2536

=.032150206

≈ 3%

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:

• The probability of Chris getting a hit is

If he comes to bat four times, what is the probability that he gets exactly 2 hits?

1

3

n Cr

prqn−r

An exactly example:

• The probability of Chris getting a hit is

If he comes to bat four times, what is the probability that he gets exactly 2 hits?

1

3

4C2

1

3

⎛

⎝⎜⎞

⎠⎟2 2

3⎛

⎝⎜⎞

⎠⎟2

=6•19•49

=2481

=827

At least example:

• The probability of Chris getting a hit is

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

3

At least example:

• The probability of Chris getting a hit is

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

3

4C3

1

3

⎛

⎝⎜⎞

⎠⎟3 2

3⎛

⎝⎜⎞

⎠⎟1

+4 C413⎛

⎝⎜⎞

⎠⎟4 2

3⎛

⎝⎜⎞

⎠⎟0

At least example:

• The probability of Chris getting a hit is

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

3

4C3

1

3

⎛

⎝⎜⎞

⎠⎟3 2

3⎛

⎝⎜⎞

⎠⎟1

+4 C413⎛

⎝⎜⎞

⎠⎟4 2

3⎛

⎝⎜⎞

⎠⎟0

=881

+181

=19

At most example:

• The probability of Chris getting a hit is

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

3

At most example:

• The probability of Chris getting a hit is

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

3

4C1

1

3

⎛

⎝⎜⎞

⎠⎟1 2

3⎛

⎝⎜⎞

⎠⎟3

+4 C013⎛

⎝⎜⎞

⎠⎟0 2

3⎛

⎝⎜⎞

⎠⎟4

=3281

+1681

=4881

=1627