Section 4-5 ProbabilitySPI 53B: compute the probability of a simple compound event

Objectives:• Find theoretical and experimental probability• Collect and analyze data for probability

Probability: • how likely something will occur

(the probability that it will rain today)

Outcome: • results of a single trial, like one roll of a number cube

Event: • any outcome or group of outcomes

Sample space: • all possible outcomes

Complement of an event: • all outcomes not in the event

Theoretical probability: • based on possible outcomes that are equally likely to occur

Experimental probability: • outcomes based on data collected

The probability of an event ranges from 0 to 1, so it will be written as a fraction, decimal, or a percent.

less likely more likely

0 0.5 1

Probability of an Event

Impossible event

Roll a 7 on a number

cube

Equally likely or unlikely

Certain to occur

Land heads or tails on a

coin

Roll a number less than 7 on a

number cube

EVENT SAMPLE SPACEFAVORABLE OUTCOMES

Apply Vocabulary

Find the probability of rolling an even number using a number cube.

Roll an even number 1, 2, 3, 4, 5, 6 2, 4, 6

outcomes possible ofnumber

outcomes favorable ofnumber y Probabilit lTheoretica

%505.2

1

6

3 number)even P(roll

Complement of an Event

Possible outcomes ofRolling a number cube

Outcomes for rollingAn even number

Complement of rollingAn even number

1, 2, 3, 4, 5, 6 2, 4, 6 1, 3, 5

P(event) + P(not event) = 1 -------- or ---------- P (not event) = 1 – P(event)

Complement of an event consists of all outcomes NOT in the event.

The probability of an event and its complement (not an event) will always equal 1.

A bowl contains 12 slips of paper, each with a different name of the month. Find the theoretical probability that a slip selected at random has a name of the month that starts with J.

Sample Space: { J F M A M J J A S O N D }

Finding Probability

outcomes possible ofnumber

outcomes favorable ofnumber y Probabilit lTheoretica

%2525.4

1

12

3

Find the complement of the event:

%7575.4

3

4

11omplement C

Experimental Probability: • based on data collected from repeated trials• P(event) = number of times an event occurs number of times the experiment is done

Experimental vs Theoretical Probability

Experimental Probability

500 belts were inspected at random. They

found no defects in 485 belts. What is the

probability that a belt selected at random

will pass quality control?

P(no defects) = number of times an event occursnumber of times the experiment is done

The probability that a belt has no defects is 97%.

= Substitute.485500

= 0.97 = 97% Simplify. Write as a percent.

If the belt manufacturer has 6258 belts, predict how many

belts are likely to have no defects. Use the .97 (no defects)

from previous example.

number with no defects = P(no defects) • number of belts

= 0.97 • 6258 Substitute. Use 0.97 for 97%.

= 6070.26 Simplify.

Approximately 6070 belts are likely to have no defects.

Probability

How does experimental probability of rolling a number cube and throwing a 2 compare to theoretical probability?

Theoretical: P(rolling a 2) = ______

Experimental: Make a chart of the results

1. P(roll 2 after 10 rolls) =

2. P(roll 2 after 20 rolls) =

3. P(roll 2 after 30 rolls) =

# of number cube rolls 10 20 30

# cube lands on 2

* The more times you roll the cube, the closer experimental should be to theoretical.

Rolling a Number Cube (Group of Two)