Chapter 4 Probability

Chapter 4 Probability

Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time)

What is probability?

Probability is a mathematical description of randomness and uncertainty. Random experiment is an experiment that produces an outcome that cannot be predicted in advance (hence the uncertainty).

The Big Picture of Statistics

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Example: coin toss

The result of any single coin toss is random.

Possible outcomes: Heads (H) Tails (T)

Coin toss

The result of any single coin toss is random. But the result over many tosses is predictable.

First series of tosses Second series

The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.

The Law of Large Numbers

As a procedure repeated again and again, the relative frequency probability of an event tends to approach the actual probability.

Spinning a coin



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Sample Space

This list of possible outcomes an a random experiment is called the sample space of the random experiment, and is denoted by the letter S.

Examples

Toss a coin once: S = {H, T}. Toss a coin twice: S = {HH, HT, TH, TT} Roll a dice: S = {1, 2, 3, 4, 5, 6} Chose a person at random and check his/her blood type: S = {A,B,AB,O}.

Sample space

Important: It's the question that determines the sample space.

A basketball player shoots three free throws. What are the possible sequences of hits (H) and misses (M)?

S = {HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM }

Note: 8 elements, 23

A basketball player shoots three free throws. What is the number of baskets made?

S = {0, 1, 2, 3}

An Event

An event is an outcome or collection of outcomes of a random experiment. Events are denoted by capital letters (other than S, which is reserved for the sample space).

Example: tossing a coin 3 times. The sample space in this case is: S = {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT} We can define the following events: Event A: "Getting no H" Event B: "Getting exactly one H" Event C: "Getting at least one H"

Example

Event A: "Getting no H" --> TTT Event B: "Getting exactly one H" --> HTT, THT, TTH Event C: "Getting at least one H" --> HTT, THT, TTH, THH, HTH, HHT, HHH

Probability

Once we define an event, we can talk about the probability of the event happening and we use the notation: P(A) - the probability that event A occurs, P(B) - the probability that event B occurs, etc.

The probability of an event tells us how likely is it for the event to occur.

Probability of an Event

0 The event is more likely to 1/2 The event is more likely to 1

NOT occur than to occur

occur than to occur

The event will NEVER occur

The event is as likely to occur as it is NOT to occur

The event will occur for CERTAIN

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Equally Likely Outcomes

If you have a list of all possible outcomes and all outcomes are equally likely, then the probability of a specific outcome is

Example: roll a die

Possible outcomes: S={1,2,3,4,5,6}

Each of these are equally likely. Event A: rolling a 2 The probability of rolling a 2 is P(A)=1/6 Event B: rolling a 5 The probability of rolling a 5 is P(A)=1/6

Example: roll a die

Event E: getting an even number.

Since 3 out of the 6 equally likely outcomes make up the event E (the outcomes {2, 4, 6}), the probability of event E is simply P(E)= 3/6 = 1/2.

Example: roll two dice

What is the probability of the outcomes summing to five?

This is S:

{(1,1), (1,2), (1,3), ......etc.}

There are 36 possible outcomes in S, all equally likely (given fair dice). Thus, the probability of any one of them is 1/36. P(sum is 5) = P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 * 1/36 = 1/9 = 0.111

A couple wants three children. What are the arrangements of boys (B) and girls (G)?

Genetics tells us that the probability that a baby is a boy or a girl is the same, 0.5.

Sample space: {BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG} All eight outcomes in the sample space are equally likely. The probability of each is thus 1/8.

A couple wants three children. What are the numbers of girls (X) they could have?

The same genetic laws apply. We can use the probabilities above to calculate the probability for each possible number of girls.

Sample space {0, 1, 2, 3} P(X = 0) = P(BBB) = 1/8 P(X = 1) = P(BBG or BGB or GBB) = P(BBG) + P(BGB) + P(GBB) = 3/8

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Probability Rules

1. The probability P(A) for any event A is 0 P(A) 1. 2. If S is the sample space in a probability model, then

P(S)=1. 3. For any event A, P(A does not occur) = 1- P(A).

Examples

Rule 1: For any event A, 0 P(A) 1

Determine which of the following numbers could represent the probability of an event?

0 1.5 -1 50% 2/3

Examples

Rule 2: P(sample space) = 1

Example

Rule 3: P(A) = 1 ? P(not A) It can be written as P(not A) = 1 ? P(A) or P(A)+P(not A) = 1

What is probability that a randomly selected person does NOT have blood type A?

P(not A) = 1 ? P(A) = 1 ? 0.42 = 0.58

Note

Rule 3: P(A) = 1 ? P(not A) It can be written as P(not A) = 1 ? P(A) or P(A)+P(not A) = 1

In some cases, when finding P(A) directly is very complicated, it might be much easier to find P(not A) and then just subtract it from 1 to get the desired P(A).

Odds

Odds against event A: P( A) P(not A)

= P( A) P( A) expressed as a:b Odds in favor event A:

P( A) P( A) =

P( A) P(not A)

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Example

Event A: rain tomorrow. The probability of rain tomorrow is 80%

What are the odds against the rain tomorrow?

P( A) = P(not A) = 0.2 = 1 = 1:4 P( A) P( A) 0.8 4

What are the odds in favor of rain tomorrow?

P( A) =

P( A)

= 0.8 = 4 = 4:1

P( A) P(not A) 0.2 1

P(A) = 0.8

Example: lottery

The odds in favor of winning a lottery is 1:1250

This means that the probability of winning is

1 = 0.0008 = 0.08% 1251

Rule 4

We are now moving to rule 4 which deals with another situation of frequent interest, finding P(A or B), the probability of one event or another occurring. In probability "OR" means either one or the other or both, and so, P(A or B) = P(event A occurs or event B occurs or both occur)

Examples

Consider the following two events: A - a randomly chosen person has blood type A, and B - a randomly chosen person has blood type B.

Since a person can only have one type of blood flowing through his or her veins, it is impossible for the events A and B to occur together.

On the other hand...Consider the following two events: A - a randomly chosen person has blood type A B - a randomly chosen person is a woman.

In this case, it is possible for events A and B to occur together.

Disjoint or Mutually Exclusive Events

Definition: Two events that cannot occur at the same time are called disjoint or mutually exclusive.

Decide if the Events are Disjoint

Event A: Randomly select a female worker. Event B: Randomly select a worker with a college

degree. Event A: Randomly select a male worker. Event B: Randomly select a worker employed part

time. Event A: Randomly select a person between 18 and 24 years old. Event B: Randomly select a person between 25 and 34 years old.

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