1. Define your events and outcomes. Probability is the likelihood of one or more events happening divided by the number of possible outcomes. So, let's say you're trying to find the likelihood of rolling a three on a six-sided die. "Rolling a three" is the event, and since we know that a six-sided die can land any one of six numbers, the number of outcomes is six. Here are two more examples to help you get oriented:[1]
Example 1: What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week?
"Choosing a day that falls on the weekend" is our event, and the number of outcomes is the total number of days in a week, seven.
Example 2: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red?
"Choosing a red marble" is our event, and the number of outcomes is the total number of marbles in the jar, 20.
2. Divide the number of events by the number of possible outcomes. This will give us the probability of a single event occurring. In the case of rolling a three on a die, the number of events is one (there's only one three on each die), and the number of outcomes is six. You can also think of this as 1 ÷ 6, 1/6, .166, or 16.6%. Here's how you find the probability of our remaining examples:[2]
Example 1: What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week?
The number of events is two (since two days out of the week are weekends), and the number of outcomes is seven. The probability is 2 ÷ 7 = 2/7 or .285 or 28.5%.
Example 2: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red?
The number of events is five (since there are five red marbles), and the number of outcomes is 20. The probability is 5 ÷ 20 = 1/4 or .25 or 25%.
4.Break the problem into parts. Calculating the probability of multiple events is a matter of breaking the problem down into separate probabilities. Here are three examples:[3]
Example 1: What is the probability of rolling two consecutive fives on a six-sided die?
You know that the probability of rolling one five is 1/6, and the probability of rolling another five with the same die is also 1/6.
These are independent events, because what you roll the first time does not affect what happens the second time; you can roll a 3, and then roll a 3 again.
Example 2:Two cards are drawn randomly from a deck of cards. What is the likelihood that both cards are clubs?
The likelihood that the first card is a club is 13/52, or 1/4. (There are 13 clubs in every deck of cards.) Now, the likelihood that the second card is a club is 12/51.
You are measuring the probability of dependent events. This is because what you do the first time affects the second; if you draw a 3 of clubs and don't put it back, there will be one less fewer club and one less card in the deck (51 instead of 52).
Example 3: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If three marbles are drawn from the jar at random, what is the probability that the first marble is red, the second marble is blue, and the third is white?
The probability that the first marble is red is 5/20, or 1/4. The probability of the second marble being blue is 4/19, since we have one fewer marble, but not one fewer blue marble. And the probability that the third marble is white is 11/18, because we've already chosen two marbles. This is another measure of a dependent event.
Multiply the probability of each event by one another. This will give you the probability of multiple events occurring one after another. Here's what you can do:[4]
Example 1:What is the probability of rolling two consecutive fives on a six-sided die? The probability of both independent events is 1/6.
This gives us 1/6 x 1/6 = 1/36 or .027 or 2.7%.
Example 2: Two cards are drawn randomly from a deck of cards. What is the likelihood that both cards are clubs?
The probability of the first event happening is 13/52. The probability of the second event happening is 12/51. The probability is 13/52 x 12/51 = 12/204 or 1/17 or 5.8%.
Example 3: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If three marbles are drawn from the jar at random, what is the probability that the first marble is red, the second marble is blue, and the third is white?
The probability of the first event is 5/20. The probability of the second event is 4/19. And the probability of the third event is 11/18. The probability is 5/20 x 4/19 x 11/18 = 44/1368 or 3.2%.