# Probability

**Probability Definition**

Probability is about determining the likelihood that an action will occur by measuring the ratio of desired outcomes to all possible outcomes. It’s one of the most obviously practical applications of math because it can help people make predictions, and it is often applied in the field of statistics.

**Probability Examples**

A simple example of probability math is predicting whether a tossed coin will land on heads or tails. The probability formula can be expressed as: number of desired outcomes/all possible outcomes

If we apply this to the coin toss, there is one desired outcome (for example, heads) out of two possible outcomes. Thus, the probability of a coin toss coming out heads is 1/2.

We can also use probability to determine the likelihood that two events will happen together. If you roll two dice, for example, you can calculate the possibility of rolling a total of 5 by first determining how many possible combinations of two dice there are. Since a die has six sides, there are a total of 36 possible combinations. How many of those equal 5? There are 4 possibilities: 1+4, 2+3, 3+2, and 4+1. Thus, the probability of rolling a 5 is 4/36 or 1/9.

**Other Probability Formulas**

Another use of probability is calculating the chances that two independent events will occur. To figure it out, we multiply the probability of the first event (A) by the probability of the second event (B), which can be expressed mathematically as P(A and B) = P(A) x P(B).

For example, to calculate the likelihood of tossing a coin twice and getting heads both times, we multiply 1/2 x 1/2 to get a probability of 1/4.

We can also calculate the probability of either of two events occurring using the following formula: P(A or B) = P(A) + P(B) – P(A and B)

For example, to figure out the likelihood of rolling 2 dice and getting either a 4 (A) or a 5 (B) or both, we first calculate that P(A) = 1/12 and P(B) = 1/9. So:

1/12 + 1/9 – (1/12 x 1/9) = 5/27

**Probability Problems**

Try the following probability math problems and see if you can put the formulas to use.

*Problem 1:* If you roll a single six-sided die, what is the probability that you will roll a 6?

*Solution:* There is one possible desired outcome out of 6 total possible outcomes, so the probability of rolling a 6 can be expressed as 1/6.

*Problem 2:* If you draw one random card from one complete deck of 52 cards and another card from another complete deck, what are the chances that you will draw a red card the first time and a queen the second time?

*Solution:* Since these are two independent events, you’ll need to calculate the probability of each draw and multiply them. The probability of selecting a red card P(A) is 26/52 or 1/2. The probability of drawing a queen P(B) is 4/52, or 1/13. Thus

1/2 x 1/13 = 1/12

*Problem 3:* If you roll a six-sided dice two times, what is the probability of rolling either a 1 on the first roll or a 6 on the second?

*Solution:* Using the formula P(A or B) = P(A) + P(B) – P(A and B), you end up with the following result:

1/6 + 1/6 – (1/6 x 1/6) = 11/36