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:

\begin{align*}P \text{(event)} = \frac{favorable \ outcomes}{total \ outcomes}\end{align*}

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

\begin{align*}P (5) &= \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{6}\\ P(\text{blue}) &= \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{4}\end{align*}

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:

\begin{align*}P \text {(green)} = \frac{favorable \ outcomes}{total \ outcomes} = \frac{3}{10}\end{align*}

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:

\begin{align*}P \text {(green)} = \frac{favorable \ outcomes}{total \ outcomes} = \frac{2}{9}\end{align*}

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 \begin{align*}R(\text{red})\end{align*} and event \begin{align*}T(\text{top})\end{align*} 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.**

\begin{align*} & \mathbf{red - top} \qquad \ \text{blue}-\text{top}\\ & \text{red}-\text{bottom} \quad \text{blue}-\text{bottom}\end{align*}

The probability of red-top is:

\begin{align*}P(\text{red}-\text{top}) = \frac{favorable \ outcomes} {total \ outcomes} = \frac{1}{4} \end{align*}

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 \begin{align*}E(\text{even})\end{align*} and event \begin{align*}S(4)\end{align*} both occurring?

**Solution**

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

\begin{align*}E (\text{even}) &= 2, \mathbf{4}, 6 \\ S(4) &= \mathbf{4} \end{align*}

**Step 2:** Find the total number of outcomes.

\begin{align*}\text{total outcomes} &= 1, 2, 3, 4, 5, 6 \\ &= 6 \ \text{total outcomes}\end{align*}

**Step 3:** Find the probability of the overlapping events.

\begin{align*}P(4) = \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{6} \end{align*}

### Video Review

### Practice

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.