# 8.11 - Probability

8.11  The student will

a)  compare and contrast the probability of independent and dependent events;

b)  determine probabilities for independent and dependent events.

### BIG IDEAS

• I can determine how many different outfits can be made from the clothes in a closet, figure my chances of making honor roll, and conclude the first, second, and third place winners of a race.
• I will understand that the probability of compound events involves finding the probability of each event and then determining how to combine them.

### UNDERSTANDING THE STANDARD

2016 VDOE Curriculum Framework - 8.11 Understanding

·  A simple event is one event (e.g., pulling one sock out of a drawer and examining the probability of getting one color).

·  If all outcomes of an event are equally likely, the theoretical probability of an event occurring is equal to the ratio of desired outcomes to the total number of possible outcomes in the sample space.

·  The probability of an event occurring can be represented as a ratio or the equivalent fraction, decimal, or percent.

·  The probability of an event occurring is a ratio between 0 and 1. A probability of zero means the event will never occur. A probability of one means the event will always occur.

·  Two events are either dependent or independent.

·  If the outcome of one event does not influence the occurrence of the other event, they are called independent. If two events are independent, then the probability of the second event does not change regardless of whether the first occurs. For example, the first roll of a number cube does not influence the second roll of the number cube. Other examples of independent events are, but not limited to: flipping two coins; spinning a spinner and rolling a number cube; flipping a coin and selecting a card; and choosing a card from a deck, replacing the card and selecting again.

·  The probability of two independent events is found by using the following formula:
P(A and B) = P(A)∙P(B)

-  Example: When rolling a six-sided number cube and flipping a coin, simultaneously, what is the probability of rolling a 3 on the cube and getting a heads on the coin?

·  If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. For example, if you choose a blue card from a set of nine different colored cards that has a total of four blue cards and you do not place that blue card back in the set before selecting a second card, the chance of selecting a blue card the second time is diminished because there are now only three blue cards remaining in the set.  Other examples of dependent events include, but are not limited to: choosing two marbles from a bag but not replacing the first after selecting it; determining the probability that it will snow and that school will be cancelled.

·  The probability of two dependent events is found by using the following formula: P(A and B) = P(A)∙P(B after A)

-  Example: You have a bag holding a blue ball, a red ball, and a yellow ball.  What is the probability of picking a blue ball out of the bag on the first pick then without replacing the blue ball in the bag, picking a red ball on the second pick?

P(blue and red) = P(blue)∙P(red after blue) =

### ESSENTIALS

• How are the probabilities of dependent and independent events similar? Different?
If events are dependent then the second event is considered only if the first event has already occurred. If events are independent, then the second event occurs regardless of whether or not the first occurs.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

·  8.11b1  Determine the probability of two independent events.

·  8.11b2  Determine the probability of two dependent events.

·  8.11a1  Determine whether two events are independent or dependent.

·  8.11a2  Compare and contrast the probability of independent and dependent events.

### KEY VOCABULARY

probability, independent event, dependent event, outcomes, replacement

Updated: Nov 20, 2018