12.7 Conditional Probability
Conditional Probability-
Example: A single card is selected from a deck of cards. Find the probability of selecting an 8,
given that it is red.
Example: Given a family with 2 children , and assuming that boys and girls are equally likely,
find the probability that the family has:
a) 2 boys
b) 2 boys if you know that at least 1 child is a boy
c) 2 boys given the older child is a girl
Conditional Probability Formula -
12.7 Practice Problems
1) If a single fair die is rolled, find the probability of a 5 given that the number rolled is odd.
2) If two fair dice are rolled, find the probability of a sum of 6 given that the roll is a "double".
3) If two cards are drawn without replacement from a deck, find the probability that the
second card is a spade, given that the first card was a spade.
4) If two cards are drawn without replacement from a deck, find the probability that the
second card is red, given that the first card was a heart.
5) If two cards are drawn without replacement from a deck, find the probability that the
second card is a face card, given that the first card was a queen.
Two marbles are drawn without replacement from a box with 3 white, 2 green, 2 red, and 1
blue marble. Find the probability.
6) The second marble is red given the first marble is white.
7) The second marble is white given the first marble is blue.
8) The second marble is blue given the first marble is red.
9) The second marble is blue given the first marble is blue.
Use the table to find the probability.
10) The following table indicates the preference for different types of soft drinks by three age
groups.
a) If a person is selected at random, find the probability that the person is over 40 years of age.
b) If a person is selected at random, find the probability that the person is over 40 and drinks
cola.
c) If a person is selected at random, find the probability that the person is over 40 years of age
given that they drink root beer.
d) If a person is selected at random, find the probability that the person drinks root beer given
that they are over 40.
12.8 Counting Principle and Permutations
Recall the Counting Principle
M*N
Example 1: License plate with 3 letters and 3 digits. How many ways can this be done if:
a) Repetition is allowed _______ _______ _______ _______ _______ _______
b) No repetition and no 0 in the 1st digit ______ ______ ______ ______ ______ ______
c) 1st letter vowel, 2nd letter consonant, digits no zeros, no repetition
_______ _______ _______ _______ _______ _______
Example 2: 6 pictures places in a row
a) How many ways can these be arranged? ______ ______ ______ ______ _____ _____
b) How many ways if 2nd and 3rd are decided to be 2 specific people in any order?
_______ _______ _______ _______ _______ _______
Permutations: any ordered arrangement of a given set of objects. Assume repetition is not
allowed.
***Formula***
Number of Permutations of n distinct items is:
n! = n (n-1)(n-2)…(3)(2)(1)
Note: 0! = 1
Example 3: In how many ways can 8 children be arranged in a line?
Example 4: What if we only want to arrange PART of a group of items?
a) Vowels {A,E,I,O,U}
Notation :
Arrange into 3 spots, no repetition.
= # of permutations of 5 items taken 3 at a time.
***Formula***
Number of Permutations when selecting part
Example 4 from above:
Example 5: 10 members of a basketball team. Randomly select a captain, 1st alternate, and
2nd alternate captain. How many choices?
____________________________________________________________________________
Permutations of Duplicate Items
Example 6: How many permutations of letters in DAD?
Permutations of Duplicate Items Formula
Example 6 from above:
Example 7: How many permutations of the letters in Mississippi?
12.9 Combinations
Permutations---order matters!
Combination- a distinct group (or set) of objects without general regard to their arrangement.
Permutation
Combination
Ex. 1 5 starters on a basketball team
1) How many ways can you choose a captain , alternate captain? (This would be a
permutation)
2) How many ways can you choose 2 starters to discuss topics with the coach?
(combination)
Formula***** Number of Combinations of n items taken r at a time
Ex. {A,E,I,O,U} select 2, order does not matter
Ex. Pres. Mojock invites 10 faculty members to lunch. There is room in his car for 6 faculty. In
how many ways can 6 faculty be chosen to ride with Pres. Mojock?
Ex. You have a test on which there are 6 questions. You are required to select any 4 questions.
How many different tests could be chosen? Order does not matter.