Now back to the Talent Show.

The Talent Show has been a huge success and the judges have had a very difficult task. There are three prizes that will be given. After a lot of deliberation, the judges have narrowed it down to the following five finalists.

2 Sixth Graders

2 Seventh Graders

1 Eighth Grader

Given these standings, what is the probability that a seventh grader and an eighth grader will be selected for an award?

**This is best solved by thinking about dependent and independent events. Pay close attention to this Concept and you will know how to figure out the probability by the end of it.**

### Guidance

Suppose you have two events:

Event \begin{align*}A\end{align*}: Spin red on spinner \begin{align*}A\end{align*}

Event \begin{align*}B\end{align*}: Spin purple on spinner \begin{align*}B\end{align*}

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

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

So:

\begin{align*}P(\text{red}) & = \frac{1}{4}\\ P (\text{purple}) & = \frac{1}{3}\end{align*}

Now a question arises. ** Does event** \begin{align*}A\end{align*}

**\begin{align*}B\end{align*}**

*affect the probability of event***That is, does the arrow landing on red in the first spinner affect the way the arrow lands in the second spinner? If not, then the two events are said to be**

*in any way?***.**

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

Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} above are independent events. No matter how many times you spin spinner \begin{align*}A\end{align*}, the outcome of spinning spinner \begin{align*}A\end{align*} does not affect the outcome of spinning spinner \begin{align*}B\end{align*}.

Look at this scenario.

Jeremy flips a coin two times. Event \begin{align*}A\end{align*} is the first coin flip. Event \begin{align*}B\end{align*} is the second coin flip. Are the two coin flips independent events?

**Ask yourself,** *Can the outcome of the first event in any way change the outcome of the second event?***If not, then the two events are independent.**

Suppose Jeremy’s first flip comes up heads. Does that in any way affect the outcome of the second flip? Is it now more likely to come up heads or tails?

**In fact, the first flip does not affect the second flip.** The probability of heads in the second flip is \begin{align*}\frac{1}{2}\end{align*}, no matter what the first flip was. Similarly, the probability of tails in the second flip is also \begin{align*}\frac{1}{2}\end{align*}, no matter what the first flip was. So **the two events are independent.**

Now that you know about ** independent events**, you can learn about

**. If one event does depend on another event, then the events depend on each other.**

*dependent events*Look at this scenario.

Mariko pulls a red sock from the laundry bag. Does this change the probability that the next sock Mariko pulls out of the bag will be red?

**Here, the act of taking a sock out of the bag changes the situation.** For the first sock, the probability of pulling out a red sock was:

\begin{align*}P(\text{red}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{3}{6} = \frac{1}{2}\end{align*}

**For the second sock, there are now only 5 socks left in the bag and only 2 of them are red. So the probability of pulling out a red sock now for the second sock is:**

\begin{align*}P(\text{red}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{2}{5}\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.*

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

*Write the difference between dependent events and independent events in your notebook.*

Try a few of these on your own. Determine whether the events described are dependent or independent events.

#### Example A

A box contains a penny, a nickel, a dime, and a quarter. What is the probability of pulling a quarter out of the box, putting it in your pocket, then pulling a penny out of the box?

**Solution: Dependent events**

#### Example B

In a laundry bag with 3 red socks and 3 blue socks, Mariko pulls out a blue sock, sees it’s the wrong sock and returns it to the bag. Now Mariko pulls out a second sock. What is the probability that it will be red?

**Solution: Independent events**

#### Example C

In a laundry bag with 3 red socks and 3 blue socks, Mariko pulls out a blue sock and keeps it out. Now Mariko pulls out a second sock. What is the probability that it will be blue?

**Solution: Dependent events**

**Here is the original problem once again. Reread it and then begin to work on figuring out the probability question.**

The Talent Show has been a huge success and the judges have had a very difficult task. There are three prizes that will be given. After a lot of deliberation, the judges have narrowed it down to the following five finalists.

2 Sixth Graders

2 Seventh Graders

1 Eighth Grader

Given these standings, what is the probability that a seventh grader and an eighth grader will be selected for an award?

**Before we even begin to figure out the probability, we first need to decide if these are independent events or dependent events.**

**Think about it this way, if we select a seventh grader for an award, then the number of possible finalists for the next award changes from 5 to 4. One event depends on the other event. Therefore, these are dependent events.**

**To figure out the probability of dependent events, we can multiply. We want to figure out the probability of a seventh and an eighth grader being selected for an award.**

\begin{align*}P(7^{th} \text{and} \ 8^{th}) = \frac{2}{5} \cdot \frac{1}{4}\end{align*}

**We have 2 out of 5 for seventh graders. Then once a seventh grader is selected, we go to four finalists. One is an eighth grader, so we have a 1 out of 4 chance of having an eighth grader for an award.**

**Next, we multiply.**

\begin{align*}P(7^{th} \text{and} \ 8^{th}) = \frac{2}{5} \cdot \frac{1}{4} = \frac{3}{20}\end{align*}

**What is this as a percent?**

\begin{align*}\frac{3}{20} = \frac{15}{100} = 15\%\end{align*}

**There is only a 15% chance of both a seventh grader and an eighth grader as award winners!**

### Vocabulary

- Independent Events
- events where one event does not impact the result of another.

- Dependent Events
- events where one event does impact the result of another.

### Guided Practice

Here is one for you to try on your own.

Does this situation describe dependent or independent events?

Kelsey has a drawer full of earrings and necklaces. She has four pairs of earrings and six necklaces in the drawer. Kelsey first takes out a necklace. She isn't happy with it, so she puts it back into the drawer. What is the probability of pulling another necklace?

**This situation describes independent events because Kelsey put the necklace back in the drawer.**

### Video Review

- This is a Khan Academy video on independent events.

- This is a Khan Academy video on dependent events.

### Practice

Directions: Write whether each pair of events is dependent or independent.

1. A: Mike rolls a number cube. B: Leah spins a red-blue-green spinner.

2. A: In a game of Go Fish, the probability of one player drawing a Queen from the deck. B: On the next player’s turn, the probability of drawing a Queen.

3. A: The probability that a randomly ordered pizza will be large. B: The probability that the same randomly ordered pizza will be deep-dish.

4. A: The probability that a randomly ordered 2-topping pizza will have pepperoni. B: The probability that the same randomly ordered 2-topping pizza will have mushrooms.

5. A: The probability of flipping a coin tails 5 times in a row. B: The probability of the sixth flip turning out to be heads.

6. A: In a 4-team league, the probability of the Rockets finishing in first place. B: In a 4-team league, the probability of the Sharks finishing in first place.

7. A: On a roll of a number cube, the probability of rolling 6. B: On a second roll of a number cube, the probability of rolling 6.

8. A: In a spelling bee, the probability of the first contestant being given the word *khaki* from a list of 10 words. B: In a spelling bee, the probability of the second contestant getting the word *khaki* from the same list of words.

9. A: The probability that it will snow on Tuesday. B: The probability that Tuesday will fall on an odd day of the month.

10. A: The probability that it will be below 32 degrees on Tuesday. B: The probability that it will snow on Tuesday.

11. A: The probability that it will snow on Tuesday. B: The probability that school will be cancelled on Tuesday.

12. A: The probability that the first Wednesday in June will fall on an even day of the month. B: The probability that the first Thursday in June will fall on an even day of the month.

13. A: The probability that the first Wednesday in June will fall on an even day of the month. B: The probability that the first Thursday in June will be sunny.

14. A: The probability that a coin will land on heads. B: The probability that a number cube will land on 5.

15. A: The probability that the first spin of a red-blue-green spinner will land on green. B: The probability that the second spin of a red-blue-green spinner will land on green.