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Chapter 4 Notes.notebook
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between 0 and 1 to indicate the likelihood of an event. We use
event is to occur.
4.1 What is Probability?
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use three major methods:
1) Intuition
3) Equally Likely Outcomes
Intuition - prediction based on previous outcomes.
Relative Frequency - we have already discussed what relative frequency is when we looked at
Probability Formula for Relative Frequency
Probability of an Event = fn
Where f is the frequency of an event, and n is the sample size.
Example: What is the probability of selecting a female student in this class?
In the long run, as the sample size increases and increases, the relative frequency of outcomes get
closer and closer to the theoretical (or actual) probability value.
An example of how the law of large numbers works is gambling at a casino.
Equally Likely Outcomes - when events have the same chance of happening.
Example: The probability of correctly guessing the answer to true-false questions.
Probability of an Event = number of outcomes favorable to an event
total number of outcomes
Can you think of any other situations where there are equally likely outcomes?
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Sample Space
results in
a definite outcome. Usually the outcome is in the form of a description, count, or
measurement.
For example: If you toss a coin, there are only 2 possible outcomes (heads or tails).
Sample Space - set of all possible outcomes.
It is especially convenient to know the sample space where all outcomes are likely
because then we can compute probabilities of various events using the following formula.
total number of outcomes
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Five Important Facts about Probability
probability is, the more likely the event is.
3) The sum of the probabilities of outcomes in a sample space is 1.
4) Probabilities can be assigned by using three methods: intuition, relative
or the formula for equally likely outcomes.
5) The probability that an event occurs plus the probability that the same event does
not occur is 1.
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1) If the probability that an event will occur is p, what is the probability that the event will not occur?a) p c) p - 1 d) 1 - p
2) If the probability that an event will occur is x/4, what is the probability that the event will not occur?a) (1 - x)/4 c) (4 - x)/x d) (4 - x)/4
3) If two fair dice are tossed once, the probability of getting 12 is 1/36. What is the probability of notgetting 12?
a) 35/36 b) 6/36 c) 30/36 d) 34/36
4) On a test the probability of getting the correct answer to a certain question is represented by x/7.Which of the following can not be a value of x?
a) -1 c) 7
5) When a number is chosen at random from the set {1,2,3,4,5,6}, which one of the following eventshas the greatest probability of occurring?
a) not choosing either 1 or 6 c) choosing a number greater than 3b) choosing an even number d) choosing a prime number
Examples
6) The probability of drawing a red marble from a sack of marbles is 2/5. Which one of the following sets ofmarbles could the sack contain?a) 4 red marbles and 6 green marbles c) 2 red marbles and 5 green marblesb) 6 red marbles and 15 green marbles d) 2 red marbles, 1 blue marble, 4 white marbles
probability that it is not green?b) 1/9 c) 5/9 d) 4/9
probability that it is not green?a) 3/5 b) 2/5 c) 4/5 d) 1/5
9) The footlights of a stage have 12 red bulbs, 8 blue bulbs, and 10 yellow bulbs. If all the bulbs areexpected to last the same amount of time, what is the probability that a yellow bulb will burn out first?
a) 20/30 b) 10/20 c) 10/30 d) 1/30
10) During a half hour of television programming, eight minutes is used for commercials. If a televisionset is turned on at a random time during the half hour, what is the probability that a commercial is not being shown?
b) 1 c) 22/30 d) 0
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Independent Events - events where the occurrence or nonoccurrence of one does not
For example: deck.
Since you replaced it, the probabilities when you select the 2nd card do not change.
the probabilities of your second selection would change.
Dependent Events - when the outcome of the first event changes the probability of
the next event.
Does the independence or dependence of an event matter?
Independence or dependence determines the way we compute probability of two
For Independent Events
For Dependent Events
" " "
Example: If you want to compute the probability of drawing an ace or a king on 2
It is important to distinguish between the "or" combinations and the "and" combinationsbecause we apply different rules to compute the probabilities.
Probability Rules:
"OR" Problems - add
"AND" Problems - multiply
4.2 Some Probability Rules- Compound Events
1) Dice2) Deck of 52 Cards
Examples:1) The probability of throwing two fours on a single toss of a pair of dice is
a) 1/6 c) 1/12 d) 1/36
2) If two coins are tossed the probability of getting two tails is
a) 1/8 c) 1/4
3) If two cards are drawn from a standard deck of 52 cards without replacement, what isthe probability that both cards are fives?
a) 4/52· 3/52 b) 5/52 · 4/51 c) 1/4 · 1/3 d) 2/52
4) From a deck of 52 cards, two cards are randomly drawn without replacement. What isthe probability of drawing two hearts?
a) 13/52 · 12/51 b) 13/52 · 13/51 c) 2/52 d) 13/52 · 13/51
5) If two cards are drawn from a standard deck of 52 cards without replacement, what is
the probability that both cards will be black aces?a) 2/52 · 2/51 b) 4/52 · 3/51 c) 4/52 · 4/51 d) 2/52 · 1/51
6) If 2 cards are dealt randomly from a standard deck of 52 cards, what is the probabilitythat they are both red queens?
a) 2/52 · 1/51 b) 2/26 c) 4/52 · 31/51 d) 2/52
7) From a standard deck of 52 cards, two cards are drawn at random without replacement.What is the probability that both cards drawn are aces?
a) 12/2,652 b) 4/2,652 c) 4/52 d) 6/2,652
probability of obtaining, at random and without replacement, two yellow gumballs?b) 36/121 c) 30/121
are chosen at random without replacement, what is the probability that both marbles will beyellow?
a) 3/14 b) 7/56 c) 1/3
drawn at random, what is the probability both are blue?a) 6/9 b) 30/72 c) 2/9 d) 30/81
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If events are dependent, the occurrence ofone event changes the probability of the
The notation P(A|B) is read “the probability
P(A, given B) equals the probability that
occurred.
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Examples
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Combination of Several EventsThe addition rule for mutually exclusive events can be extended so that it applies to
the situation in which we have more than two events that are mutually exclusive to all other events.
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Summary
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2. Multiplication Rule of Counting
3. Permutations
4. Combinations
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a method of listing outcomes of
an experiment consisting of a series of activities
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possible outcomes for event E possible outcomes for event E then there are a total ofpossible outcomes for the series of events E
This rule extends to outcomes created by a series of three, four, or more events. Just simply multiply the number ofevents to get the total number of outcomes for the series.
Factorial NotationFor any counting number ( ),
! = ( – 1)( – 2)…1
0! = 1
1! = 1
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Examples
Evaluate 4!
Evaluate 5!
Evaluate 6!
Evaluate 7!
Jean is making sandwiches for a class picnic. She is using 4 different fillings with 2 different kinds of bread. Howmany different kinds of sandwiches can she make using one kind of filling on one kind of bread for each sandwich?
On a restaurant menu, there are six sandwich choices and three beverage choices. How many different lunchesmay a person order consisting of one sandwich and one beverage?
John has 6 pairs of pants and 3 shirts. How many possible outfits consisting of one shirt and one pair of pants canhe select?
alternative CD's, 1 country CD, 2 Jazz CD's, and 2 Pop CD's from which to choose. How many differentcombinations of CD's can be played?
stripes, and all the other items are solid colors. If Josh will not wear stripes and checkered patterns together, howmany different shirt and pants combinations can Josh wear?
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Permutations:
Permutation
Permutations are especially useful when the order of the data is important.
Therefore, we need to enter the data in this order:
n n
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Examples
1. Evaluate 7P3
2. Evaluate 4P3
3. Evaluate 9P2
4. Evaluate 8P4
5. Evaluate 10P3
6. How many different 4-letter arrangements can be formed using the letters of the word “JUMP”, ifeach letter is used only once?
7. How many different five-digit numbers can be formed from the digits 1,2,3,4, and 5 if each digit isused only once?
8. How many different 6-letter arrangements can be formed using the letters in the word “ABSENT”,if each letter is used only once?
9. All seven-digit telephone numbers in a town begin with 245. How many telephone numbers maybe assigned in the town if the last four digits do not begin or end in a zero?
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Combinations:
Combinations
In combination problems, order is not taken into consideration. Therefore, the difference between permutations andcombinations is that in permutations we are considering groupings and in combinations we are considering only the number of
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Examples
1. Evaluate 7C3
2. Evaluate 9C3
3. Evaluate 10C2
4. Evaluate 8C6
5. Evaluate 4C3
6. Find the number of combinations of 6 things taken 3 at a time.
7. How many different committees of 3 people can be chosen from a group of 9people?
8.positions individual play while making the first selection, how many teams canbe formed if 14 candidates try out and the coach selects 5 players?
9.of 22 songs can be selected?
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