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# 12.16: Independent Events

Difficulty Level: At Grade Created by: CK-12
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Practice Independent Events
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Miranda is on the school tennis team and she is waiting for the medals to be given out after the district tournament. There are three medals - 1st, 2nd, and 3rd place - that are awarded one at a time. She is standing with two of her teammates and there are five players from other schools.  The 3rd and 2nd place medals are given to the players from other schools.   Is the awarding of the medals an independent event?

In this concept, you will learn about independent events.

### Guidance

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

Suppose you have two events:

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

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

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

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

So:

P(red)P(purple)=14=13\begin{align*}P(\text{red}) & = \frac{1}{4}\\ P (\text{purple}) & = \frac{1}{3}\end{align*}

Now a question arises:  Does event A\begin{align*}A\end{align*} affect the probability of event B\begin{align*}B\end{align*} in any way? 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 independent events.

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

Let's look at an example.

Jeremy flips a coin two times. Event A\begin{align*}A\end{align*} is the first coin flip. Event B\begin{align*}B\end{align*} is the second coin flip. Are the two coin flips independent events?  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 12\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 12\begin{align*}\frac{1}{2}\end{align*}, no matter what the first flip was. So the two events are independent.

Here's another example.

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:

P(red)=favorable outcomestotal outcomes=36=12\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:

P(red)=favorable outcomestotal outcomes=25\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.

### Guided Practice

Does the following situation describe 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?

First, evaluate the 1st outcome:

Kelsey takes a necklace.  She replaces it.

Next, evaluate the 2nd outcome:

Kelsey will be taking a piece of jewelry from the same collection as she did for the 1st outccome.

Then, decide if the 1st outcome affects the 2nd outcome:

Since Kelsey replaced the necklace she took, the 1st outcome does not affect the 2nd outcome.

The answer is the two events are independent.

### Examples

Determine whether or not the events described are independent events.

#### Example 1

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?

First, evaluate the 1st outcome:

You pull a quarter out of a box and placed in your pocket.

Next, evaluate the 2nd outcome:

You will pull another coin from the box.

Then, decide if the 1st outcome affects the 2nd outcome:

Since you did not place the quarter back into the box, there is one less coin to pull from for the 2nd outcome

The answer is the two events are not independent.  The 1st coin pull affects the outcome of the 2nd coin pull.

#### Example 2

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?

First, evaluate the 1st outcome:

Mariko pulls a blue sock out of the laundry bag and replaces it.

Next, evaluate the 2nd outcome:

Mariko will pull another sock out of the laundry bag.

Then, decide if the 1st outcome affects the 2nd outcome:

Since Mariko replaced the sock he pulled out of the laundry bag for the 1st outcome, the same socks remain in the laundry bag for the 2nd outcome.

The answer is the two events are independent.  Taking the sock from the bag for the 1st outcome does not affect taking a sock from the bag for the 2nd outcome since both pulls contain the same socks.

#### Example 3

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?

First, evaluate the 1st outcome:

Mariko takes a sock from the laundry bag and keeps it.

Next, evaluate the 2nd outcome:

He will then pull another sock.

Then, decide if the 1st outcome affects the 2nd outcome:

Since Mariko did not replace the sock on the 1st pull, there is one less sock to pull from on the 2nd pull.

The answer is the two events are not independent because there is one less sock to pull from on the 2nd pull.

Remember Miranda waiting to find out if she will receive the 1st-place medal?  There are three medals that are awarded one at a time for 1st place, 2nd place, and 3rd place.  The 2nd and 3rd-place medals have already been awarded to players from another school.  Is the awarding of the 1st-place medal an independent event?

First, evaluate the 1st and 2nd outcomes:

The 3rd-place medal was awarded to a player from the other school.

The 2nd-place medal was then awarded to another player from the other school.

Next, evaluate the 3rd outcome:

The 1st-place medal will be awarded to one of the remaining players, Miranda, one of her teammates, or the three remaining players from the other school.

Then, decide if the 1st and 2nd outcomes affect the 3rd outcome:

Each time a medal is presented, it is no longer available to the remaining players.  There is only 1 medal left for the 3rd outcome.

The answer is the events are not independent.

### Explore More

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.

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