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?

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

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

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

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Now a question arises.
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Does event A affect the probability of event B in any way?
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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
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independent events
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.

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Definition:
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If the outcome of one event has no effect on the outcome of a second event, then the two events are
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independent events.
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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.
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Now let’s think about a different kind of example, one where the outcome of one event does impact the outcome of another event.
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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?

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Solution
:
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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:

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:

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
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dependent events
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.

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Definition:
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If the outcome of one event has an effect on the outcome of a second event, then the two events are
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dependent events.
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Sometimes, we have mutually exclusive events and we have events that overlap and are not mutually exclusive.

Events
and event
are
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overlapping events
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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.

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To calculate the probability of overlapping events, list the sample space and find the favorable events.
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The probability of red-top is:

Are the following events independent or dependent events?

#### Example A

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

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Solution: Independent events.
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#### 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?

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Solution: Independent events
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#### Example C

Why?

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Solution: Because the blue marble was replaced, it does not affect the outcome of drawing of red.
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Now let's go back to the dilemma from the beginning of the Concept.

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The runners are independent. The speed of one runner does not impact the speed of another runner.
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### 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 and event both occurring?

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Solution
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Step 1
:
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Identify the overlapping outcomes of both events.

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Step 2
:
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Find the total number of outcomes.

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Step 3
:
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Find the probability of the overlapping events.

### Video Review

### Explore More

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

- A: Doug flips a coin. B: Marlene chooses a card out of a deck.
- 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.
- A: Eddie chooses the color blue for his new bike. B: Eddie chooses lasagne from the dinner menu.
- A: The probability that it will rain tomorrow. B: The probability that the Red Wings hockey team will win their game tomorrow.
- A: The probability that it will rain tomorrow. B: The probability that the baseball team will have a rain delay.
- 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.
- 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.
- 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.
- A: The probability that it will snow tomorrow. B: The probability of having a snow day from school.
- A: The probability that it will be 90 degrees. B: The probability of enjoying a hot day at the beach.
- A: The probability that it will rain tomorrow. B: The probability of getting an A on a math test.
- A: The probability that the Rockies will be in the playoffs. B: The probability that the Rockies will win the World Series.
- A: The probability that tomorrow will be sunny. B: The probability that tomorrow will be a full moon.
- A The probability that tomorrow will be sunny. B: The probability that tomorrow will be cloudy.
- A: The probability that it will cold today. B: The probability that it will be a full moon tomorrow.