Section_04_01 - it

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SECTION 4.1

BASIC IDEAS

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Objectives

1.

2.

3.

Construct sample spaces Compute and interpret probabilities Approximate probabilities using the Empirical Method Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Objective 1

Construct sample spaces

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Probability Experiment

A

probability experiment

is one in which we do not know what any individual outcome will be, but we do know how a long series of repetitions will come out.

For example, if we toss a fair coin, we do not know what the outcome of a single toss will be, but we do know what the outcome of a long series of tosses will be – about half “heads” and half “tails”.

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Probability

The

probability

of an event is the proportion of times that the event occurs in the long run. So, for a “fair” coin, that is, one that is equally likely to come up heads as tails, the probability of heads is 1/2 and the probability of tails is 1/2.

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Law of Large Numbers

The law of large numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability.

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Sample Space

The collection of all the possible outcomes of a probability experiment is called a

sample space

.

Example:

 Suppose that a coin is tossed. The sample space consists of:

{Heads, Tails}

 Suppose that a standard die is rolled. The sample space consists of:

{1, 2, 3, 4, 5, 6}

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Event

We are often concerned with occurrences that consist of several outcomes. For example, when rolling a die, we might be concerned with the possibility of rolling an odd number. A collection of outcomes of a sample space is called an

event

.

Example:

A probability experiment consists of rolling a die. The sample space is {1, 2, 3, 4, 5, 6}.

The event of rolling an odd number = {1, 3, 5}.

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Probability Model

Once we have a sample space for an experiment, we need to specify the probability of each event. This is done with a

probability model

. We use the letter “P” to denote probabilities.

For example, if we toss a coin, we denote the probability that the coin lands heads by “P(Heads).”

Notation:

If A denotes an event, the probability of event A is denoted by P(A).

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Probabilities With Equally Likely Outcomes If a sample space has n equally likely outcomes, and an event A has k outcomes, then

 Number of outcomes in

A

Number of outcomes in the sample space 

k n

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Objective 2

Compute and interpret probabilities

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Example

A fair die is rolled. Find the probability that an odd number comes up.

Solution:

The sample space has six equally likely outcomes: {1, 2, 3, 4, 5, 6} The event of an odd number has three outcomes: {1, 3, 5} The probability is:

P

(Odd Number)   6 1 2 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example

A family has three children. Denoting a boy by B and a girl by G, we can denote the genders of these children from oldest to youngest. For example, GBG means the oldest child is a girl, the middle child is a boy, and the youngest child is a girl. There are eight possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. Assume these outcomes are equally likely. What is the probability that all three children are the same gender?

Solution:

Of the eight equally likely outcomes, the two outcomes BBB and GGG correspond to having all children of the same gender. Therefore

P

(All three have same gender)   8 1 4 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Probability Rules

The probability of an event is always between 0 and 1. That is, 0 ≤ P(A) ≤ 1. If A cannot occur, then P(A) = 0.

If A is certain to occur, then P(A) = 1.

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Sampling From a Population

Sampling an individual from a population is a probability experiment. The population is the sample space and members of the population are equally likely outcomes.

Example:

There are 10,000 families in a certain town categorized as follows: Own a house Own a condo Rent a house Rent an apartment 4753 1478 912 2857 A pollster samples a single family from this population. What is the probability that the sampled family rents?

Solution:

The number of families who rent is 912 + 2857 = 3769. Therefore, the probability that the sampled family rents is 3769/10,000 = 0.3769.

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Unusual Events

An

unusual event

is one that is not likely to happen. In other words, an event whose probability is small.

A rule of thumb is that any event whose probability is less than 0.05 is considered to be unusual. Example:

In a college of 5000 students, 150 are math majors. A student is selected at random and turns out to be a math major. Is this an unusual event?

Solution:

The event of choosing a math major consists of 150 students out of a total of 5000 students. The probability of choosing a math major is 150/5000 = 0.03. Since 0.03 < 0.05, this would be considered an unusual event.

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Objective 3

Approximate probabilities using the Empirical Method

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Approximating Probabilities with the Empirical Method

The law of large numbers says that if we repeat a probability experiment a large number of times, then the proportion of times that a particular outcome occurs is likely to be close to the true probability of the outcome. The

Empirical Method

consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome.

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Example

The Centers for Disease Control reports that in the year 2002 there were 2,057,979 boys and 1,963,747 girls born in the U.S. Approximate the probability that a newborn baby is a boy.

Solution:

The number of times that the experiment has been repeated is: 2,057,979 boys + 1,963,747 girls = 4,021,726 births The proportion of births that are boys is: 2,057,979/4,021,726 = 0.5117

Therefore, the probability that a newborn baby is a boy is approximated by 0.5117.

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Do You Know…

• • • • How to construct a sample space?

How to compute probabilities of equally likely events?

The rules of probability?

How to compute probabilities using the Empirical Method?

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