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Aim: Final Review Session 3
Probability
HW15: worksheet
Exam 5/9 and 5/11
Three Rules
• Sometimes we need to know all possible
outcomes for a sequence of events
– We use three rules
1. Fundamental counting rule
2. Permutation rule
3. Combination rule
Factorial Notation
• Uses exclamation point
– 5! = 5 *4 * 3 * 2 * 1
• For any counting n: n! = n(n-1)(n-2)(n-3)…1
• 0! = 1
Permutations
• Permutations: an arrangement of n objects in
a specific order
• The arrangement of n objects in a specific
order using r objects at a time is called a
permutation of n objects taking r objects at
atime
– It is written as nPr and the formula is
n!
n Pr 
(n  r )!
Combinations
• Combinations: a selection of distinct objects
without regard to order
• The number of combinations of r objects
selected from n objects is denoted by nCr and
is given by the formula
n!
n Cr 
(n  r )!r !
Basic Concepts
• Probability: the chance of an event occurring
• Probability Experiment: is the chance process
that leads to well-defined results called
outcomes.
• Outcomes: is the result of a single trial of
probability experiments
• Sample Space: the set of all possible outcomes of
a probability experiment
• Event: consists of a set of outcomes of a
probability experiment
Four Basic Probability Rules
1. The probability of any event E is a number (either
a fraction or a decimal) between and including 0
and 1. This is denoted by 0  P ( E )  1
2. If an event E cannot occur, it is a probability of 0
3. If an event E is certain, then the probability of E is
1
4. The sum of the probability of all the outcomes in
the sample space is 1
What is the two-part description of
the probability model?
1. A list of possible outcomes
2. A probability for each outcome
First part of probability model
1. A probability model first tells us what outcomes
are possible
- The sample space S of a random phenomenon is
the set of all possible outcomes.
-
The name “sample space” is natural in random
sampling, where each possible outcome is a sample
and the sample space contains all possible samples.
To specify S, we must state what constitutes an
individual outcome and then state which outcomes
can occur.
Sample space can be either qualitative or categorical
Example of Sample Space (S)
• Toss a coin. There are only two possible
outcomes, and the sample space is
or, more briefly, S = {H, T}.
*No many how many times or how many coins are
tossed, the sample size will always be H and T.
Second part of probability model
2. A probability for each outcome
-
An event is an outcome or a set of
outcomes of a random
phenomenon.
- That is, an event is a subset of
the sample space.
Example of an Event
• Take the sample space S for four tosses of a
coin to be the 16 possible outcomes in the
form HTTH. Then “exactly 2 heads” is an
event. Call this event A. The event A expressed
as a set of outcomes is
In a probability model, events have
probabilities
1.
2.
3.
4.
Any probability is a number between 0 and 1. An event with probability
0 never occurs, and an event with probability 1 occurs on every trial. An
event with probability 0.5 occurs in half the trials in the long run.
All possible outcomes together must have probability 1. Because every
trial will produce an outcome, the sum of the probabilities for all
possible outcomes must be exactly 1.
If two events have no outcomes in common, the probability that one or
the other occurs is the sum of their individual probabilities. If one event
occurs in 40% of all trials, a different event occurs in 25% of all trials, and
the two can never occur together, then one or the other occurs on 65%
of all trials because 40% + 25% = 65%.
The probability that an event does not occur is 1 minus the probability
that the event does occur. If an event occurs in (say) 70% of all trials, it
fails to occur in the other 30%. The probability that an event occurs and
the probability that it does not occur always add to 100%, or 1.
What are the probability rules?
• We use capital letters near the beginning of the alphabet to denote events
• If A is any event, we write its probability as P(A).
Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.
Rule 2. If S is the sample space in a probability model, then P(S) = 1.
Rule 3. Two events A and B are disjoint if they have no outcomes in common
and so can never occur together. If A and B are disjoint,
– This is the addition rule for disjoint events.
Rule 4. The complement of any event A is the event that A does not occur,
written as Ac. The complement rule states that
Disjoint Events = Never Happen =
Nothing in Common
What is the probability that an accident occurs on a
weekend, that is, Saturday or Sunday? Because an
accident can occur on Saturday or Sunday but it
cannot occur on both days of the week, these two
events are disjoint. Using Rule 3, we find
ANS: The chance that an accident occurs
on a Saturday or Sunday is 5%.
Complement = Has Everything Else
Suppose we want to find the probability
that a phone-related accident occurs on
a weekday.
-To solve this problem, we could use
Rule 3 and add the probabilities for
Monday, Tuesday, Wednesday,
Thursday, and Friday. However, it is
easier to use the probability that we
already calculated for weekends and
Rule 4.
-The event that the accident occurs on
a weekday is the complement of the
event that the accident occurs on a
weekend. Using our notation for
events, we have
What is a complement of an event?
• The complement of an event E is the et of
outcomes in the sample space that are not
included in the outcomes of event E. The
complement of E is denoted by
• Example: Find the complement of Eeach event
1. Rolling a die and getting a 4. (Ans: Getting a 1,
2, 3, 5, 6)
2. Selecting a letter of the alphabet and getting a
vowel. (Ans: Getting a consonant – assuming y
is a constant)
The Rule for Complementary Events
P( E )  1  P( E )
 P( E )  P( E )  1
Example: If the probability that a person lives in an industrialized
country of the world is 1/5, find the probability that a person
does not live in an industrialized country.
Ans: P(not living in an industrialized country) = 1 – P(living in industrialized country)
= 1 – 1/5
= 4/5
“And” / “Or”
• In probability:
– And = subtract
– Or = Add
Difference between two examples
• Mutually exclusive events: two events are
mutually exclusive if they cannot occur at the
same time (they have no outcomes in
common)
– One cannot be a Democrat and an Independent at
the same time; must be one or the other
Binomial Probability
What is a success? What is a failure?
• Successful Trials: (p) getting the wanted
outcome
• Failure Trials: (1-p=q) getting the unwanted
outcome
Binomial Probability
• In general for a given experiment, if the
probability of success in p and the probability
of failure is 1 - p = q, then the probability of
exactly r successes in n independent trials is
r
n
Cr p q
nr
Binominal Probability: “Exactly”
• If a fair coin is tossed 10 times, what is the
probability that it falls tails exactly 6 times?
– Procedure:
• (1) Find the probability of getting the wanted outcome
– P(tails) = 1/2
• (2) Find the probability of getting the unwanted
outcome
– P(not tails) = 1/2
• Put it into the formula
6
4
 1   1  105
10 C6 
   
 2   2  512
Binomial Probability: “At Least”
• A coin is loaded so that the probability of heads is 4 times the probability
of tails. What is the probability of at least 1 tail in 5 throws?
– Procedure:
• (1) probability of success
– P(tails) = 1/5
• (2) probability of failure
– P(not tails) = 4/5
• (3) at least = that number up to the max
– P(at least 1 tail in 5) = P(1) + P(2) + P(3) + P(4) + P(5)
• (4) plug in formula and calculate
1
4
2
3
3
2
4
1
5
1 4
1 4
1 4
1 4
1 4
C

C

C

C

C
5 1  
 5 2    5 3    5 4    5 5   
5 5
5 5
5 5
5 5
5 5
2101

3125
0
Binomial Probability: “at most”
• A family of 5 children is chosen at random. What is the probability that
there are at most 2 boys in this family of 5?
– Procedure:
• (1) probability of success
– P(boys) = 1/2
• (2) probability of failure
– P(not boy) = 1/2
• (3) at most = that number down to 0/min
– P(at most 2 boys in 5) = P(2) + P(1) = P(0)
• (4) plug in formula and calculate
2
3
1
4
0
1 1
1 1
1 1
C

C

C
5 2
   5 1    5 0   
2
  2
2 2
 2  2
1

2
5