Sample Spaces and Probability

CHAPTER 4.1

“Life is a school of probability” ~ Walter Bagehot

“The only two sure things are death and taxes” ~ cynical philosopher

“Statistically, the probability of any one of us being here is so small that you'd think the mere fact of existing would keep us all in a contented dazzlement of surprise” ~ Lewis Thomas

PROBABILITY

Probability can be defined as the chance of an event occurring

A probability experiment is a chance process that leads to well-defined results called outcomes

An outcome is the result of a single trial of a probability experiment

A sample space is the set of all possible outcomes of a probability experiment

PROBABILITY

Experiment Sample Space

Toss one coin  

Roll a die  

Answer a true/false question  

Toss two coins  

EXAMPLES OF SAMPLE SPACE AND PROBABILITY EXPERIMENTS

Find the sample space for rolling two dice

SAMPLE SPACE

Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl.

SAMPLE SPACE

A tree diagram is a device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment

TREE DIAGRAM

MAKE A TREE DIAGRAM THE FAMILY WITH 3 CHILDREN

An event consists of a set of outcomes of a probability experiment

An event with one outcome is called a simple event

A compound event consists of two or more outcomes or simple events

EVENT

1. Classical or theoretical

2. Empirical or Relative frequency or Experimental

3. Subjective

THE THREE TYPES OF PROBABILITY

Classical probability uses sample spaces to determine the numerical probability that an event will happen. You do not actually perform the experiment to determine the probability.

Classical probability assumes that all outcomes in the sample space are equally likely to occur.

CLASSICAL PROBABILITY

The probability of any event E is the number of outcomes in E divided by the total number of outcomes in the sample space.

Denoted by

FORMULA FOR CLASSICAL PROBABILITY

)(

)()(

Sn

EnEP

What is the probability that the sun will rise tomorrow?

What is the probability that Indiana Jones will come crashing through our window?

What is the probability of getting heads when a coin is flipped?

RANGE OF VALUES FOR PROBABILITY

Probabilities should be expressed as reduced fractions or rounded to two or three decimal places.

When the probability of an event is an extremely small decimal, round to the first nonzero digit after the decimal point.

For example: 0.0000587 would be rounded to 0.00006

ROUNDING RULES

Equally likely events are events that have the same probability of occurring.

For example, flipping a coin and getting heads or tails

EQUALLY LIKELY EVENTS

1. P(3)

2. P(even)

3. P(odd)

4. P(prime)

5. P(4 or 5)

6. P(value less than 7)

FIND THE PROBABILITY OF EACH EVENT USING A SINGLE DIE

1. P(jack)

2. P(heart)

3. P(black ten)

4. P(six of clubs)

5. P(3 or 6)

6. P(3 or diamond)

FIND THE PROBABILITY FOR EACH EVENT USING A STANDARD DECK OF

CARDS

If a family had three children, find the probability that exactly two of the three children are girls.

Find the probability that at least one of the three children is a boy.

GENDER OF CHILDREN