Aim: Final Review Session 3Probability

HW15: worksheetExam 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 rule2. Permutation rule3. 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 r

nP

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 r

nC

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

2. If an event E cannot occur, it is a probability of 03. If an event E is certain, then the probability of E

is 14. The sum of the probability of all the outcomes in

the sample space is 1

0 ( ) 1P E

What is the two-part description of the probability model?

1. A list of possible outcomes2. 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. 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.

2. 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.

3. 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%.

4. 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 each event1. 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)

E

The Rule for Complementary Events

( ) 1 ( )

( ) ( ) 1

P E P E

P E P E

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 rn rC p q

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 formula6 4

10 6

1 1 105

2 2 512C

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 calculate1 4 2 3 3 2 4 1 5 0

5 1 5 2 5 3 5 4 5 5

1 4 1 4 1 4 1 4 1 4

5 5 5 5 5 5 5 5 5 5

2101

3125

C C C C C

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 calculate2 3 1 4 0 5

5 2 5 1 5 0

1 1 1 1 1 1

2 2 2 2 2 2

1

2

C C C