Question

Two dice are thrown together. Find the probability of getting a sum equal to 8.

Answer 1

Hint: To find the probability of any event, we will apply the formula given by a Favorable number of outcomes divided by the total number of outcomes. Here, a favorable number of outcomes means that the number of outcomes that is considered for the probability.

Complete step-by-step answer:
According to the question, two dice are chosen here numbered as 1, 2, 3, 4, 5, and 6 on each of its faces respectively. Since two dice are thrown together so all the probabilities are represented as a sample space denoted by 5. Therefore,
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Now, we will choose those ordered pairs of dice which after summed up gives the result equal to 8. Therefore, we get the favorable event as (2, 6), (3, 5), (4, 4), (5, 3), (6, 2).
Now, we will count the elements in the sample space. Thus, we have a total number of events equal to 36. Also, we will count the favorable events. These are 5. Now, we will apply the formula of probability given as,
\[\text{Probability = }\dfrac{\text{Favorable Number of Outcomes}}{\text{Total number of outcomes}}\]
Here, the number is represented as a short form of the number. Therefore, we have
\[\text{Probability = }\dfrac{5}{36}\]
Now, as we can see, the terms 5 and 36 do not have any common terms, so we cannot cancel it further to any simpler forms. Therefore this is the simplest form in which we can express the fraction of the probability. We can also express it as 0.139 in decimal form.
Hence, the required probability is given by \[\dfrac{5}{36}\] or 0.139.

Note: The probability of any event is always between 0 and 1. In this question also the probability is 0.139 which is between the numbers 0 and 1. If in the case, any probability of an event comes as a negative number or greater, then it will be considered as the wrong solution. Alternatively, we could have tried to find out the probability of those events in which the sum of the pair of the dice does not imply the number 8. And then apply the formula of probability given by P(E) + P(E’) = 1 where P(E) is the probability of pairs of those numbers which results in 8 after summing up and P(E’) is the probability of pairs of those numbers which results in not 8. Then, we could have found P(E) = 1 - P(E’). But it will be a long method.