Probability (Single events)

Probabilities are fractions that represent how often something should happen (if the event is random).  We can talk about probabilities in fractions or percents, but they are really just fractions that represent how often a particular outcome should appear (numerator of fraction) out of the total number of outcomes (denominator of fraction).

Think about rolling a die:

• A die has 6 sides, each with a different number. 
• When you roll a die, the numbers turn up randomly. 
• So, the probability of any particular number coming up is 1 out of 6 or $\dfrac{1}{6}$.  
• (Remember, probabilities are theoretical - they represent what should happen in a random world.  We all know that randomness is not totally predictable, especially in a small sample.  If you rolled some dice, you might get the same number 3 times in a row or you might not get a number for 10 throws, but if you roll enough times, theoretical probabilities come close).

So, you might get a problem that asks you to find the the probability of a certain event.  You find the frequency of the event you are interested in, put it on the top of the fraction (numerator), then find the total number of events and put it on the bottom of the fraction (denominator).  Reduce the fraction if possible.  To find the probability as a percent, divide the numerator by the denominator:

For example: 

You are going to roll a die.  What is the probability that you will get a 3?

The total number of outcomes you are interested in (rolling a 3) = 1

The total possible number of out comes (1, 2, 3, 4, 5, or 6) = 6

Write the fraction:

$\dfrac{\text{Outcome you care about}}{\text{Total possible outcomes}}=\dfrac{1}{6}$

This fraction cannot be reduced but it can be written as a percent:

$\dfrac{1}{6}=1\div6=.1\overline{66}$

So if you roll a die one time, the probability of rolling a 3 is $\dfrac{1}{6}$ or 16.67%.

We use probabilities all the time -- and not just for gambling and other types of games.  Sometimes people want to know the probability that they will get chosen for something.  Let's say you are in the audience for a game show, and you want to figure out your chances of getting called. Assuming the choosing process is random (it may not be!), you calculate your probability in the same way:

$\dfrac{\text{Thing you care about}}{\text{Total number of things}}$

For example: 

You are in the audience for a game show.  There are 230 people in the audience.  The show will call up 10 people at random. What are your odds of being chosen?

The total number of outcomes you are interested in (number of people who will be chosen) = 10

The total number of outcomes (all the people) = 230

Write the fraction:

$\dfrac{\text{Outcome you care about}}{\text{Total outcomes}}=\dfrac{10}{230}$

That's a clunky fraction, so simplify:

$\dfrac{10}{230}=\dfrac{1}{23}$

Or, write it as a percent:

$\dfrac{1}{23}=1\div23=.043478$

So, if audience members are chosen at random, your chance of being chosen is  $\dfrac{1}{23}$ or 4.35%.

Other times, calculating probabilities require thinking about subsets of groups.  But, the general process remains the same.  The top of the fraction is the number you care about, and the bottom fraction is the total number of possibilities.

For example: 

Your school is choosing a spokesperson to speak at a national conference.  The school has already chosen the top 50 students at the school. Five (5) of the candidates are first years, 16 are sophomores, 13 are juniors, and 16 are seniors.  If the final spokesperson will be chosen at random from among these candidates, you can calculate the probability that the spokesperson will be from a particular class by creating a fraction.  Let's say you want to find the probability that the spokesperson will be a junior.

The total number of outcomes you are interested in (juniors) = 13

The total number of outcomes (all the candidates) = 50

Write the fraction:

$\dfrac{\text{Outcome you care about}}{\text{Total outcomes}}=\dfrac{13}{50}$

That's a clunky fraction, but it can't be simplified, so a percent will be a clearer answer here:

$\dfrac{13}{50}=13\div50=.26$

So, if the spokesperson is chosen at random from among the candidates, the likelihood that the candidate will be a junior is 26%.

A note about notation: probability is typically written as "P."  A question asking about the probability of rolling a three, might ask "What is P(3)?"  If you're doing statistics problems and see notation like "P(3)" or "P(green)" or "P(A)" don't think you're supposed to multiply.  The "P" is just asking you for the probability of the event in the parentheses that follow the P.

Overall, you calculate the probability of a single event occurring by creating a fraction, with the event(s) you are interested in on top, and the total number of events (or possibilities) on the bottom.  Then, treat the fraction like any other fraction to express your probability clearly.