Two fair dice are rolled. What is the probability that the sum of the numbers rolled is eight?
Any fair die has six faces numbered from one to six. We want the sum or total of our two numbers to equal eight. One way of answering this question is to set up a two-way table. Each of the two die could land on any number from one to six. For example, they could both land on one. This would give a sum or total of two. If the first die landed on one and the second die on two, we would have a sum of three. We can complete the table showing all the possibilities as shown.
The probability of any event occurring can be written as a fraction, the number of successful outcomes over the number of possible outcomes. In this question, a successful outcome would be a sum of eight. There are five ways this could occur: rolling a two and a six, a three and a five, two fours, a five and a three, and a six and a two. This means the number of successful outcomes is five.
There are 36 possible outcomes as there are 36 numbers in the table; six multiplied by six is 36. When two fair dice are rolled, the probability that the sum of the numbers is eight is five out of 36.
An alternative method here would be to list the successful outcomes or combinations: two and six, three and five, four and four, five and three, and six and two. As there are five of these, we know that the correct probability is five out of 36.