# Probability (Using Probability)

Probabilities show the chances (in percent or fraction form) of something happening (if that event is happening at random). Probabilities are usually given in simplest form. So, the odds of rolling a 5 on a die is $\dfrac{1}{6}$.

We can use these probabliies to predict how often something SHOULD happen if you do it a certain number of times. We can also use probabilities to work backwards and figure out actual numbers in the scenario you have a probability for.

In the lesson about Probabilities (Single event), we learned that the probability of rolling a 3 is $\dfrac{1}{6}$. But you might want to know how mant 3s you will get if you roll the die 100 times, or a thousand times. You find that number by multiplying your probability by the number of times you'll roll the die.

*Example*:

*How many times will you get 3 if you roll the die 100 times? *

Multiply 100 times $\dfrac{1}{6}$ and you'll get your answer:

$100 \times \dfrac{1}{6}=16.67$

In 100 rolls, you should roll 3 about 17 times.

In the lesson about Probabilities (Single event), we also talked about a game show: we decided that an audience member's chances of being called up in that particular game show was $\dfrac{1}{23}$. So, if someone were to attend that game show 25 times, how many times should be be called?

*Example*:

*If your odds of being called up in a game show are $\dfrac{1}{23}$ how many times should you be called up if you attend 25 shows? *

Multiply 25 times $\dfrac{1}{23}$ and you'll get your answer:

$100 \times \dfrac{1}{23}=1.087$

If you attend 25 games, you should get called up once.

In the lesson about Probabilities (Single event), we also talked about the probability that a school's spokesperson would come from a specific grade. We can also use the probability we found to figure out how many candidates belonged to a certain grade.

*Example*:

*A national spokesperson will be chosen for your school. 50 qualified candidates have been selected and the spokesperson will be drawn at random from among these candidates. If there is a 26% chance that the spokesperson will be a junior, how many candidates were juniors? *

The probability that a junior will be chose is calculated by:

$\dfrac{\text{# we're interested in}}{\text{Total #}}=\dfrac{\text{Juniors}}{\text{All candidates}}=\dfrac{x}{50}=.26$

From there, you just have to work backwards:

$$\eqalign{\dfrac{x}{50}&=.26\\\times 50 \quad &\quad \times 50\\x&=13}$$

There were 13 juniors among the candidates.

You can always work backwards from probabilities to find actual numbers. Let's try one more.

*Example*:

*A national spokesperson will be chosen for your schools. 50 qualified candidates have been selected and the spokesperson will be drawn at random from among these candidates. If there is a 84% chance that the spokesperson will be a female, how many candidates are female? *

The probability that a junior will be chose is calculated by:

$\dfrac{\text{# we're interested in}}{\text{Total #}}=\dfrac{\text{Females}}{\text{All candidates}}=\dfrac{x}{50}=.84$

From there, you just have to work backwards:

$$\eqalign{\dfrac{x}{50}&=.84\\\times 50 \quad &\quad \times 50\\x&=42}$$

There were 42 females among the candidates.

In the end, probabilities are very helpful. They can help you make predictions and they can help you work backwards to find the actual numbers that led to the probabilities. Remember, although we often present probabilities as percents, it's often helpful to think of them as fractions: the thing you care about, over the total possible outcomes.

#### Practice Problems:

## Using Probability

- You throw two dice (6-sixed number cubes), what is the probability of getting two 5s?
- You throw two dice (6-sixed number cubes), what is the probability of getting one 6 and one 1?
- You throw two dice (6-sixed number cubes), what is the probability of your two throws adding up to 12?
- You reach into your sock drawer, which contains 5 black socks, 4 white socks, and 3 blue socks. If you pull out two socks, what is the probabilty that you get a pair of blue socks?
- If you reach into the same (full) sock drawer, what are the odds of getting a pair of white socks?
- If you reach into the same (full) sock drawer, what are the olds that you pull any pair of socks?
- You want to do a magic trick, but you're not very good at it. You want your audience member to pull a Queen of hearts out of a standard deck (52 cards). If the audience member pulls 5 cards, how would you find the probability that she will pull a Queen of hearts (assuming there is no magic at work?). (Don't do the problem, just write the problem you'd want to do!)
- Using a deck of cards, what is the probability that you pull any queen?
- If you pull two cards out of a deck of cards, what are the odds that you pull any queen and then any ace?
- If a student's probability of randomly being chosen "Star of the Week" is 1/11, how many students could be in the class? What is another amount of students that might be in the class?

#### Answer Key: