A pair of fair 6-sided dice is rolled. What is the probability that a 3 is rolled if it is known that the sum of the numbers landing uppermost is less than or equal to 7? (Enter your answer as an exact fraction.)
I assume that on of one die = 3. The other numbers to get ≤ 7 are 1, 2, 3, 4. Since the other die has six sides, what are the odds?
I assume that one die = 3. The other numbers to get ≤ 7 are 1, 2, 3, 4. Since the other die has six sides, what are the odds?
To solve this problem, we need to find the probability of rolling a 3 given that the sum of the numbers is less than or equal to 7.
We can begin by listing all the possible outcomes when rolling two fair 6-sided dice. There are a total of 36 outcomes, as each die has 6 possible outcomes:
(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)
Next, we can identify the outcomes where the sum is less than or equal to 7:
(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2),
(4, 1), (4, 3), (5, 1), (6, 1), (3, 3), (2, 4), (1, 5), (1, 6), (4, 2),
(2, 5), (5, 2), (2, 6), (6, 2), (5, 3), (3, 4), (4, 4), (4, 5), (5, 4),
(5, 5), (3, 5), (6, 3), (3, 6), (6, 4), (6, 5), (4, 6), (5, 6), (6, 6)
Now, we can count how many outcomes involve rolling a 3 in the pairs:
(1, 2), (2, 1), (3, 1), (2, 4), (1, 5), (1, 6), (4, 2),
(2, 5), (5, 2), (2, 6), (6, 2), (3, 4), (4, 4), (4, 5), (5, 4),
(5, 5), (3, 5), (6, 3), (3, 6), (6, 4), (6, 5), (4, 6), (5, 6), (6, 6)
We see that there are a total of 24 outcomes that involve rolling a 3 given that the sum is less than or equal to 7.
Now, we can calculate the probability by dividing the number of outcomes with a 3 by the total number of outcomes with a sum less than or equal to 7:
P(rolling a 3 | sum less than or equal to 7) = Number of outcomes with a 3 / Total number of outcomes
P(rolling a 3 | sum less than or equal to 7) = 24 / 36
Simplifying the fraction, we get:
P(rolling a 3 | sum less than or equal to 7) = 2 / 3
Therefore, the probability of rolling a 3 given that the sum of the numbers is less than or equal to 7 is 2/3.