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# Independent Events

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Practice Independent Events
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Recognize Independent and Dependent Events

Have you ever thought about how one event can affect another event?

Do you know the difference between a dependent event and an independent event? If someone runs a 5 minute mile and someone else runs a 10 minute mile are these two events dependent on each other?

Pay attention and you will learn all about independent and dependent events in this Concept.

### Guidance

You have been learning all about probability. Now we can think about different events and how these events impact each other. Take a look at this situation.

Suppose you have two events:

Event A: Toss 5 on the number cube

Event B: Spin blue on the spinner

The probability of each of these events by itself is easy enough to compute. In general:

$P \text{(event)} = \frac{favorable \ outcomes}{total \ outcomes}$

If this is the case, then we can write the following ratios for rolling a 5.

$P (5) &= \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{6}\\P(\text{blue}) &= \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{4}$

These two events were performed with a spinner and a number cube.

Now a question arises.

Does event A affect the probability of event B in any way?

That is, does the number cube landing on 5 affect where the arrow lands in the spinner? If not, then the two events are said to be independent events .

Definition: If the outcome of one event has no effect on the outcome of a second event, then the two events are independent events.

Events A and B above are independent events. No matter how the number cube turns up, its outcome does not affect the outcome of spinning the spinner.

Now let’s think about a different kind of example, one where the outcome of one event does impact the outcome of another event.

A bag has 3 red marbles, 4 blue marbles, and 3 green marbles. Irina pulls 1 green marble out of the bag. Does this change the probability that the next marble Irina pulls out of the bag will be green?

Solution : Here, the act of taking a marble out of the bag changes the situation. For the first marble, the probability of pulling out a green marble was:

$P \text {(green)} = \frac{favorable \ outcomes}{total \ outcomes} = \frac{3}{10}$

For the second marble, there are now only 9 marbles left in the bag and only 2 of them are green. So the probability of pulling out a green marble for the second marble is now:

$P \text {(green)} = \frac{favorable \ outcomes}{total \ outcomes} = \frac{2}{9}$

Clearly, the first event affected the outcome of the second event in this situation. So the two events are NOT independent. In other words, they are dependent events .

Definition: If the outcome of one event has an effect on the outcome of a second event, then the two events are dependent events.

Sometimes, we have mutually exclusive events and we have events that overlap and are not mutually exclusive.

Events $R(\text{red})$ and event $T(\text{top})$ are overlapping events because both events share one outcome – red-top. The Venn diagram for overlapping events shows that the two events overlap, or share 1 or more outcomes.

To calculate the probability of overlapping events, list the sample space and find the favorable events.

$& \mathbf{red - top} \qquad \ \text{blue}-\text{top}\\& \text{red}-\text{bottom} \quad \text{blue}-\text{bottom}$

The probability of red-top is:

$P(\text{red}-\text{top}) = \frac{favorable \ outcomes} {total \ outcomes} = \frac{1}{4}$

Are the following events independent or dependent events?

#### Example A

Rolling a 1 and then rolling a 7 on a number cube.

Solution: Independent events.

#### Example B

A bag has a red and two blue marbles. First, one blue marble is drawn and then replaced. Then a red one is drawn. Are these two events dependent or independent?

Solution: Independent events

#### Example C

Why?

Solution: Because the blue marble was replaced, it does not affect the outcome of drawing of red.

Now let's go back to the dilemma from the beginning of the Concept.

The runners are independent. The speed of one runner does not impact the speed of another runner.

### Vocabulary

Independent Events
The outcome of one event has no effect on the outcome of a second event.
Dependent Events
If the outcome of one event has an effect on the outcome of another event they are dependent events.
Overlapping Events
Events that share one outcome

### Guided Practice

Here is one for you to try on your own.

For a single toss of a number cube, what is the probability of event $E(\text{even})$ and event $S(4)$ both occurring?

Solution

Step 1 : Identify the overlapping outcomes of both events.

$E (\text{even}) &= 2, \mathbf{4}, 6 \\S(4) &= \mathbf{4}$

Step 2 : Find the total number of outcomes.

$\text{total outcomes} &= 1, 2, 3, 4, 5, 6 \\&= 6 \ \text{total outcomes}$

Step 3 : Find the probability of the overlapping events.

$P(4) = \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{6}$

### Practice

Directions: Write whether events A and B are dependent or independent.

1. A: Doug flips a coin. B: Marlene chooses a card out of a deck.
2. A: In a bag with 5 white marbles and 5 black marbles, Sanjay pulls out a white marble. B: Without returning the marble to the bag, Sanjay pulls out a second marble.
3. A: Eddie chooses the color blue for his new bike. B: Eddie chooses lasagne from the dinner menu.
4. A: The probability that it will rain tomorrow. B: The probability that the Red Wings hockey team will win their game tomorrow.
5. A: The probability that it will rain tomorrow. B: The probability that the baseball team will have a rain delay.
6. A: From a deck of cards, the probability of one player drawing a heart from the deck. B: On the next player’s turn, the probability of drawing another heart.
7. A: The probability of a spinner landing on blue 6 times in a row. B: The probability of the spinner landing on blue on the next spin.
8. A: The probability of flipping a coin and having it come up heads. B: The probability of flipping it again and having it come up heads.
9. A: The probability that it will snow tomorrow. B: The probability of having a snow day from school.
10. A: The probability that it will be 90 degrees. B: The probability of enjoying a hot day at the beach.
11. A: The probability that it will rain tomorrow. B: The probability of getting an A on a math test.
12. A: The probability that the Rockies will be in the playoffs. B: The probability that the Rockies will win the World Series.
13. A: The probability that tomorrow will be sunny. B: The probability that tomorrow will be a full moon.
14. A The probability that tomorrow will be sunny. B: The probability that tomorrow will be cloudy.
15. A: The probability that it will cold today. B: The probability that it will be a full moon tomorrow.