In a simultaneous throw of two fair dice, what is the probability of getting a total (sum of the two faces) equal to 7?

Introduction / Context:
This is a classic probability question about rolling two fair dice. We are interested in the sum of the numbers on the two dice, and specifically we want the probability that this sum equals 7.

Given Data / Assumptions:

• Two fair six sided dice are thrown together.
• Each die shows an integer from 1 to 6.
• All 36 ordered pairs of outcomes are equally likely.
• We require those pairs whose sum is exactly 7.

Concept / Approach:
The total number of ordered outcomes is 6 * 6 = 36. To find the probability that the sum is 7, we list all pairs (first die, second die) that add up to 7, count them, and divide by 36. This is straightforward counting of favourable outcomes versus total outcomes.

Step-by-Step Solution:
Total outcomes when two dice are rolled = 6 * 6 = 36.We need pairs (a, b) such that a + b = 7, with a and b each between 1 and 6.List such pairs: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).There are 6 favourable ordered pairs.Therefore, probability that the sum is 7 = number of favourable outcomes / total outcomes = 6 / 36.Simplify 6 / 36 by dividing numerator and denominator by 6 to get 1 / 6.

Verification / Alternative check:
We can check that the pairs listed are all possible solutions to a + b = 7 with 1 ≤ a, b ≤ 6. There is no other pair that satisfies the equation within the allowed range. Also, note that the number of favourable outcomes is symmetric around (3.5, 3.5), which is consistent for a typical die sum distribution where 7 is the most likely sum.

Why Other Options Are Wrong:
The probability 1/4 would require 9 favourable outcomes among 36, which is too many. The value 1/20 would require fewer than 2 favourable outcomes, which is impossible because we have 6. The value 3/4 is far too large and would suggest that most outcomes give a sum of 7. Only 1/6 aligns with the correct count of 6 favourable outcomes out of 36 total outcomes.

Common Pitfalls:
A common error is to forget that the dice are distinguishable, so (1, 6) and (6, 1) are different ordered outcomes even though both give the sum 7. Another pitfall is miscounting the number of valid pairs by skipping one or adding an invalid combination. Carefully writing out all pairs that add to 7 helps avoid mistakes and reinforces understanding of how sums distribute in two dice rolls.

Final Answer:
The probability that the total is 7 is 1/6.