Two events are disjoint if they have no outcomes in common. Consider two events and that are disjoint. Can you say whether or not events and are independent?

#### Watch This

http://www.youtube.com/watch?v=tfOaGhp4L3A Brightstorm: Probability of Independent Events

#### Guidance

Recall that the
**
probability
**
of an event is the
**
chance
**
of it happening. Probabilities can be written as fractions or decimals between 0 and 1, or as percents between 0% and 100%. If all outcomes in an experiment have an equal chance of occurring, to find the probability of an event find the number of outcomes in the event and divide by the number of outcomes in the sample space.

Consider the experiment of flipping a coin two times and recording the sequence of heads and tails. The sample space is , which contains four outcomes. Let be the event that heads comes up exactly once. Therefore,

To find the probability of a single event, all you need to do is count the number of outcomes in the event and the number of outcomes in the sample space. Probability calculations become more complex when you consider the combined probability of two or more events.

Two events are
**
independent
**
if one event occurring does not change the probability of the second event occurring. Two events are
**
dependent
**
if one event occurring causes the probability of the second event to go up or down.
**
Two events are independent if the probability of
and
occurring together is the product of their individual probabilities:
**

**
if and only if
and
are independent events.
**

In some cases it is pretty clear whether or not two events are independent. In other cases, it is not at all obvious. You can always test if two events are independent by checking to see if their probabilities satisfy the relationship above.

**
Example A
**

Consider the experiment of flipping a coin two times and recording the sequence of heads and tails. The sample space is , which contains four outcomes. Let be the event that the first coin is a heads. Let be the event that the second coin is a tails.

a) List the outcomes in events and .

b) Take a guess at whether or not you think the two events are independent.

c) Find and .

d) Find . Are the two events independent?

**
Solution:
**

a)
.
. Note that the outcomes in the two events overlap. This
*
does NOT mean that the events are not independent!
*

b) If you get heads on the first coin, that shouldn't have any effect on whether you get tails for the second coin. It makes sense that the events should be independent.

c) .

d) is the event of getting heads first and tails second. .

Because , the events are independent.

**
Example B
**

Consider the experiment of flipping a coin two times and recording the sequence of heads and tails. The sample space is , which contains four outcomes. Let be the event that the first coin is a heads. Let be the event that both coins are heads.

a) List the outcomes in events and .

b) Take a guess at whether or not you think the two events are independent.

c) Find and .

d) Find . Are the two events independent?

**
Solution:
**

a) . .

b) If you get heads on the first coin, then you are more likely to end up with two heads than if you didn't know anything about the first coin. It seems like the events should NOT be independent.

c) . .

d) is the event of getting heads first and both heads. This is the same as the event of getting both heads, since if you got both heads then you definitely got heads first. .

Because , the events are NOT independent.

**
Example C
**

Consider the experiment of flipping a coin two times and recording the sequence of heads and tails. The sample space is , which contains four outcomes. Let be the event that both coins are heads. Let be the event that both coins are tails.

a) List the outcomes in events and .

b) Take a guess at whether or not you think the two events are independent.

c) Find and .

d) Find . Are the two events independent?

**
Solution:
**

a) . .

b) If you get both heads, then you definitely didn't get both tails. It seems like the events should NOT be independent.

c) . .

d) is the event of getting two heads and two tails. This is impossible to do because these two events are disjoint. .

Because , the events are NOT independent.

**
Concept Problem Revisited
**

If and are disjoint, then and .

For the two events to be independent, This means that . By the zero product property, the only way for is if or .

In other words, two disjoint events are independent if and only if the probability of at least one of the events is 0.

#### Vocabulary

An
**
experiment
**
is an occurrence with a result that can be observed.

An
**
outcome
**
of an experiment is one possible result of the experiment.

The
**
sample space
**
for an experiment is the set of all possible outcomes of the experiment.

An
**
event
**
for an experiment is a subset of the sample space containing outcomes that you are interested in (sometimes called

**).**

*favorable outcomes*
The
**
complement of an event
**
is the event that includes all outcomes in the sample space not in the original event. The symbol for complement is ′.

The
**
union
**
of two events is the event that includes all outcomes that are in either or both of the original events. The symbol for union is
.

The
**
intersection
**
of two events is the event that includes all outcomes that are in both of the original events

**.**The symbol for intersection is .

A
**
Venn diagram
**
is a way to visualize sample spaces, events, and outcomes.

The
**
probability
**
of an event is the chance of the event occurring.

Two events are
**
independent
**
if one event occurring does not change the probability of the second event occurring.

**if and only if and are independent events.**

Two events are
**
dependent
**
if one event occurring causes the probability of the second event to go up or down.

Two events are
**
disjoint
**

**(**if they do not have any outcomes in common.

*mutually exclusive*)#### Guided Practice

1. Consider the experiment of tossing a coin and then rolling a die. Event is getting a tails on the coin. Event is getting an even number on the die. Are the two events independent? Justify your answer using probabilities.

2. Consider the experiment of rolling a pair of dice. Event is a sum that is even and event is both numbers are greater than 4. Are the two events independent? Justify your answer using probabilities.

3. and . If , are events and independent?

**
Answers:
**

1. The sample space for the experiment is . Next consider the outcomes in the events. . . .

The events are independent because .

2. First find the sample space and the outcomes in each event:

Next find the probabilities:

The events are independent because .

3. Events and are independent if and only if .

, so the events are NOT independent.

#### Practice

Consider the experiment of flipping a coin three times and recording the sequence of heads and tails. The sample space is , which contains eight outcomes. Let be the event that exactly two coins are heads. Let be the event that all coins are the same. Let be the event that at least one coin is heads. Let be the event that all coins are tails.

1. List the outcomes in each of the four events. Which of the two events are complements?

2. Find .

3. Find . Are events and independent? Explain.

4. Find . Are events and independent? Explain.

5. Find . Are events and independent? Explain.

6. Create two new events related to this experiment that are independent. Justify why they are independent using probabilities.

Consider the experiment of drawing a card from a deck. The sample space is the 52 cards in a standard deck. Let be the event that the card is red. Let be the event that the card is a spade. Let be the event that the card is a 4. Let be the event that the card is a diamond.

7. Describe the outcomes in each of the four events.

8. Find .

9. Find . Are events and independent? Explain.

10. Find . Are events and independent? Explain.

11. Find . Are events and independent? Explain.

12. and . If are events and independent?

13. and . If are events and independent?

14. What is the difference between disjoint and independent events?

15. Two events are disjoint, and both have nonzero probabilities. Can you say whether the events are independent or not?