Probability Basics

Probability of an Event             

The probability of an event is the number of ways it can occur, divided by the total number of possible outcomes. Probability is always a number between 0 (no outcomes lead to the event) and 1(all outcomes lead to the event.) Probability can be expressed as a percent from 0 to 100. For example, when you flip a coin, the probability of getting a head is 1 (it can only occur one way) divided by 2 (total possible outcomes), or 1/2. This may also be expressed as a 50% probability.

Dependent and Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other. Coin tosses and the roll of dice are examples of independent events.

Two events are dependent if the outcome or occurrence of the first changes the probability of the second. For example, the odds of picking an ace from a shuffled deck of 52 playing cards are 4/52 or 1/13. If the card is not replaced, the odds of picking a second ace are changed to 3/51. The odds of picking any other particular card are also affected, because the total number of possible outcomes has changed from 52 to 51.

Mutually Exclusive Events

Mutually exclusive events are events that can’t happen simultaneously, such as getting both heads and tails on the same coin toss. When two events are mutually exclusive, the probability that one or the other will occur is the sum of the probability of each event. For example, when you roll a six-sided die, the probability of getting either a 3 or a 4 is the sum of the probability of each, or 1/6 + 1/6= 2/6 or 1/3.

Probability of Multiple Events

To find the probability of two independent events that occur in sequence, multiply the probabilities of the events occurring separately. For example, the odds of tossing three heads in a row are ½ x ½ x ½ or 1/8.

Events vs. Outcomes

When you roll a six-sided die, there are six equally-possible outcomes. Rolling the number four, for example, is an event that corresponds to the outcome of 4, and therefore has a 1/6 chance of occurring.

You can combine outcomes to define different events with different probabilities. For example, when rolling a six-sided die, an even number is an event that corresponds to three outcomes (rolling 2, 4 or 6). Therefore the probability is 3/6 or 50%.

Complement of an Event

The complement of an event is the set of all outcomes that do not lead to the event. If you know the probability of an event, you can find the probability of its complement by subtracting the probability of the event from 1. For example, given that the probability of rolling a 5 is 1/6, the probability of its complement – rolling 1, 2, 3, 4 or 6 – is equal to 1-1/6 or 5/6.