Download Probability—number of favorable outcomes. The likelihood that a

Probability—number of favorable outcomes.
total number of outcomes
The likelihood that a specific event will occur
Probability is a number that is in the range of 0 < x < 1
Probability experiment—an action or trial through which specific results are obtained.
Probability can be expressed as a ratio (fraction in lowest terms), a percentage
Basic Properties of probability
1. The probability of an event happening is always between 0 and 1 inclusive
2. The probability of an event does not occur is zero.
3. The probability that the event does occur is 1 (certain event, sure thing)
4. The sum of all the probabilities of all possible outcomes is 1.
5. The probability that an event does not occur is (1 minus the probability that the event
does occur.
Sample Space—the collection of all possible outcomes for an experiment
Outcome-- The result of a single probability experiment
Event—a collection of outcomes of the experiment. It is a subset of the sample space. It may
consist of one or more outcomes
Notation: “E” is an event
P(E) is the probability that the event E occurs.
P(E) = Number of way that “E” can happen
Number of all possibilities
Theoretical probability --When each outcome in a sample space is equally likely to occur.
Rolling a fair die would give a 1/6 probability for each outcome.
P(E) = Number of outcomes in event E
Total number of outcomes in sample space
Empirical probability- based on the observations obtained from a probability experiment.
P (E) = Frequency of event
Total Frequency
Translating Inequalities—when examining probabilities, make sure you know what the words
mean:
At least --this means greater than or equal to (>)
At most—means you can go up to the number in, but not beyond it . This would be less
than or equal t (<)
Not more than—means you can have the number in question or less --less than or equal
to. (<)
Not less than—means greater than or equal to (>)
Less than --you don’t include the number itself (<), just all the numbers less than the
number (<)
greater than –you don’t include the number itself, just all number bigger than the
number (>)
Empty set—null set –if the event or subset of a sample does not have any outcomes in it. You
would just make { } or 
Contingency Table—a table that classifies outcomes and their probabilities into rows and
columns.
S = { 1, 2, 3, 4, 5, 6, 7, 8}
A = {1,3, 5}
B = {2, 4, 5}
C = {4, 6}
Putting sample spaces together:
Union: To put two sets together into one (possibly) larger set. It is A
So in the above sample S with subsets A and B: A
Notice I only represented the 5 once in the union set.
B.
B = {1, 2, 3, 4, 5}
Intersection: find the common (what they share) outcomes between the two sets:
A  B = {5}
A  C = { } or 
Complements- The complement of an event A, is the set of all outcomes from the sample set S
which are not in subset A. It is denoted Ac
For the above set, Ac = { 2, 4, 6}
The union of A and A complement is the sample space S. So P(A) + P(Ac) = 1.
P(Ac) = 1 – P(A)
Conditional Probabilities: The probability of one event given that another event has already
occurred. We write it P(A|B). This means the probability of A given B. This is given that event
B has happened, what is the probability of A happening.
You use the multiplication rule to find the probability of
Example. Given you rolled a die and got an odd number, what is the probability that the
roll is a 5?
P(die is a 5| die is odd). Since the die is odd, we have three possibilities, each of which
are equally likely to occur. Therefore, it is 1/3, or .33, or 33%.
This would be the formula P(A|B) = P(A  B)
P (B)
This would take the intersection of A and B (A  B) and then divide it by the
probability of B.
Another way to think of it.
P(A|B) = P(A  B) .
P (B)
So you can cross-multiply and you get :
P(B) * P(A|B) = P(A  B)
Suppose an individual applying to a college determines that he has an 80% chance of being
accepted, and he knows that dormitory housing will only be provided for 60% of all of the
accepted students. The chance of the student being accepted and receiving dormitory housing is
defined by
Disjoint—(mutually exclusive)—have no outcomes in common and can never occur
simultaneously. If A and B are disjoint:
P(A or B) = P (A) + P (B)
Mutually Exclusive Events
Experiment 1:
A single card is chosen at random from a standard
deck of 52 playing cards. What is the probability of
choosing a 5 or a king?
Possibilities:
1. The card chosen can be a 5.
2. The card chosen can be a king.
Experiment 2:
A single card is chosen at random from a standard
deck of 52 playing cards. What is the probability of
choosing a club or a king?
Possibilities:
1. The card chosen can be a club.
2. The card chosen can be a king.
3. The card chosen can be a king and a club (i.e., the
Unit 6
king of clubs).
In Experiment 1, the card chosen can be a five or a king, but not both at the same time.
These events are mutually exclusive. In Experiment 2, the card chosen can be a club, or a
king, or both at the same time. These events are not mutually exclusive.
Definition:
Two events are mutually exclusive if they cannot occur at the same time (i.e.,
they have no outcomes in common).
Experiment 3:
A single 6-sided die is rolled. What is
the probability of rolling an odd
number or an even number?
Possibilities:
1. The number rolled can be an odd
number.
2. The number rolled can be an even
number.
Events:
These events are mutually exclusive
since they cannot occur at the same
time.
Experiment 4:
A single 6-sided die is rolled. What is the probability of rolling a
5 or an odd number?
Possibilities:
1. The number rolled can be a 5.
2. The number rolled can be an odd number (1, 3 or 5).
3. The number rolled can be a 5 and odd.
Events:
These events are not mutually exclusive since they can occur at the same
time.
Experiment 5:
A single letter is chosen at random from the word SCHOOL. What is the
probability of choosing an S or an O?
Possibilities:
1. The letter chosen can be an S
2. The letter chosen can be an O.
Events:
These events are mutually exclusive since they cannot occur at the same
time.
Experiment 6:
A single letter is chosen at random from the word SCHOOL. What is the
probability of choosing an O or a vowel?
Possibilities:
1. The letter chosen can be an O
2. The letter chosen can be a vowel.
3. The letter chosen can be an O and a vowel.
Events:
These events are not mutually exclusive since they can occur at the same
time.
Independent—Two events are independent if knowing that the one event occurs does not
change the probability that the other occurs. If A and B are independent, then
P(A and B) = P(A) * P(B) This is called the multiplication rule for independent events
Disjoint events are not independent
Two events, A and B, are independent if the fact that A occurs does not affect the probability of
B occurring.
Some other examples of independent events are:




Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
Choosing a marble from a jar AND landing on heads after tossing a coin.
Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second
card.
Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.
To find the probability of two independent events that occur in sequence, find the
probability of each event occurring separately, and then multiply the probabilities. This
multiplication rule is defined symbolically below. Note that multiplication is represented by
AND.
Experiment 1: A dresser drawer contains one pair of socks of
each of the following colors: blue, brown, red,
white and black. Each pair is folded together in
matching pairs. You reach into the sock drawer
and choose a pair of socks without looking. The
first pair you pull out is red -the wrong color. You
replace this pair and choose another pair. What is
the probability that you will choose the red pair of
socks twice?
Probabilities:
P(red)
=
1
5
P(red and red) = P(red) · P(red)
=
1
5
·
1
5
=
1
25
Experiment 2: A coin is tossed and a single 6-sided die is rolled.
Find the probability of landing on the head side of
the coin and rolling a 3 on the die.
Probabilities:
P(head)
=
1
2
P(3)
=
1
6
P(head and 3) = P(head) · P(3)
=
1
2
=
1
12
·
1
6
Experiment 3: A card is chosen at random from a deck of 52 cards.
It is then replaced and a second card is chosen.
What is the probability of choosing a jack and an
eight?
Probabilities:
P(jack)
=
4
52
P(8)
=
4
52
P(jack and 8) = P(jack) · P(8)
=
4
52
=
16
2704
=
1
169
·
4
52
Experiment 4: A jar contains 3 red, 5 green, 2 blue and 6 yellow
marbles. A marble is chosen at random from the jar.
After replacing it, a second marble is chosen. What is
the probability of choosing a green and a yellow
marble?
Probabilities:
P(green)
=
5
16
P(yellow)
=
6
16
P(green and yellow) = P(green) · P(yellow)
=
5
16
=
30
256
·
6
16
15
128
Two events are dependent if the outcome or occurrence of the first affects the outcome or
occurrence of the second so that the probability is changed.
=
=
4
52
P(jack on 2nd pick given queen on 1st pick) =
4
51
P(queen and jack)
4 4
16
4
·
=
=
52 51 2652 663
P(queen on first pick)
=
Experiment 2: Mr. Parietti needs two students to help him with a science
demonstration for his class of 18 girls and 12 boys. He
randomly chooses one student who comes to the front of the
room. He then chooses a second student from those still
seated. What is the probability that both students chosen
are girls?
Probabilities:
P(Girl 1 and Girl 2) = P(Girl 1) and P(Girl 2|Girl 1)
=
18
30
=
306
870
=
51
145
·
17
29
Experiment 3:
In a shipment of 20 computers, 3 are defective.
Three computers are randomly selected and tested.
What is the probability that all three are defective if
the first and second ones are not replaced after
being tested?
Probabilities:
P(3 defectives) =
3 2 1
6
1
·
·
=
=
20 19 18 6840
1140
Experiment 4:
Four cards are chosen at random from a deck of 52
cards without replacement. What is the probability of
choosing a ten, a nine, an eight and a seven in order?
Probabilities:
P(10 and 9 and 8 and 7) =
4 4 4 4
256
32
·
·
·
=
=
52 51 50 49 6,497,400 812,175
Experiment 5:
Three cards are chosen at random from a deck of 52
cards without replacement. What is the probability of
choosing 3 aces?
Probabilities:
P(3 aces) =
4 3 2
24
1
·
·
=
=
52 51 50 132,600 5,525