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.