Probability: Simple and Compound Independent and Dependent

Experimental and Theoretical

Understanding Probability

Probability is a measure of how likely an event is to occur. The probability of an event occurring is the ratio of the number of favorable outcomes to the number of possible outcomes. In a probability experiment, favorable outcomes are the outcomes that you are interested in.

Understanding Probability

Probability will help you decide how often something is likely to occur, but it will never help you to know exactly when that event will happen unless the probability is 0 (it will never happen) or 1 (it will always happen.)

The probability can be expressed as a fraction, a decimal, or a percent

If the probability of an event is 0, it is impossible.

If the probability of an event is 1, it is certain.

The more unlikely an event is, the closer its probability is to 0. The more likely an event is, the closer its probability is to 1.

The probability, P, of an event occurring is from 0 to 1.

If an event is impossible, its probability is 0.

If an event is certain to occur, its probability is 1.

Understanding how to find the probability of events

The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.

number of favorable outcomes

number of possible outcomes

P(event) =

A simple event in a probability experiment is determined by the outcome of one trial in the experiment.

The probability of a compound event is the combination of at least two simple events.

Understanding how to find the probability of compound events

One way to find the probability of compound events is to multiply the probabilities of the simple events that make up the compound event. If P(A) represents the probability of event A and P(B) represents the probability of event B, then the probability of the compound event (A and B ) can be represented algebraically.

P(A and B) = P(A) • P(B)

Example

1. What is the probability of drawing a white marble from a bag containing 3 red marbles and 1 white marble?

( )P white

2. What is the probability of tossing “head” twice on two fair coins tossed at the same time.

( ) ( ) ( )P H and H P H P H

Understanding the difference between dependent and independent events

If the outcome of the first event does not affect the possible outcomes of the second event, the events are called independent events.

If two events are independent, you can use this formula to find the probability of both events occurring.

P(A and B) = P(A) • P(B)

In a compound event, if the outcome of the first event affects the possible outcomes of the second event, the events are called dependent events. Probability (B given A) means the likelihood that B will happen when A happens. If two events are dependent, you can use this formula to find the probability of both events occurring.

P(A and B) = P(A) • P(B given A)

Practice

1. A bag contains 10 tiles numbered 1 through 10. One tile is drawn from the bag. What is the probability of drawing an even number?

2. You are playing Concentration and need a 6 to win. These cards are the only ones left and they are face down.

What is the probability you will pick a 6?

3. A probability experiment consists of rolling a fair number cube numbered 1 through 6 and then spinning a spinner with two equally likely outcomes, red or blue. Find the probability of rolling a 2 on the number cube and spinning red on the spinner.

4. A bag contains 6 blue marbles, 4 red marbles, and 2 green marbles. One marble is drawn from the bag, and its color is recorded. Another marble is drawn, and its color is also recorded. What is the probability of drawing 2 blue marbles if the first marble is not returned to the bag before the second marble is drawn?

5. A bag contains 10 marbles: 3 red, 5 blue, and 2 orange. Wendy draws a marble from the bag, records the color, and then puts the marble back in the bag. She repeats this for a total of 100 trials.

Theoretical probability: P(blue) = 0.50

Experimental probability: P(blue) = 0.52

6. A local bookstore surveyed to determine what type of books its customers prefer. The table below shows the results of the survey.

Based on the survey, if the bookstore plans to order 400 new books, about how many drama books should they order? Show your work.