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 6?
To solve this problem, let's start by determining the possible outcomes when rolling a pair of fair 6-sided dice.
When rolling two 6-sided dice, each die can land on any number from 1 to 6. The total number of outcomes is given by multiplying the number of outcomes for each die.
Total outcomes = Number of outcomes for Die 1 * Number of outcomes for Die 2
Since each die has 6 possible outcomes, the total number of outcomes is 6 * 6 = 36.
Now, we need to determine the favorable outcomes, which are the outcomes where the sum of the numbers landing uppermost on the two dice is less than or equal to 6.
To simplify this, we can list all the possible pairs of numbers (Die 1, Die 2) that satisfy this condition:
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)
(3, 1), (3, 2), (3, 3), (3, 4)
(4, 1), (4, 2), (4, 3)
(5, 1), (5, 2)
(6, 1)
Counting the number of favorable outcomes, we find that there are 21 pairs that satisfy the condition.
Therefore, the probability of rolling a 3 given that the sum of the numbers is less than or equal to 6 is:
Probability = Number of favorable outcomes / Total number of outcomes = 21 / 36 = 7 / 12.
So, the probability that a 3 is rolled, knowing that the sum of the numbers landing uppermost is less than or equal to 6, is 7/12.
To find the probability of rolling a 3 given that the sum of the numbers is less than or equal to 6, we first need to determine all the possible outcomes where the sum of the numbers is less than or equal to 6.
Let's consider all the possible sums of two dice rolls:
Sum 2: (1, 1)
Sum 3: (1, 2), (2, 1)
Sum 4: (1, 3), (2, 2), (3, 1)
Sum 5: (1, 4), (2, 3), (3, 2), (4, 1)
Sum 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)
Out of these possible outcomes, we need to find the ones where a 3 is rolled. From the list above, we can see that there are two outcomes where a 3 is rolled: (1, 2) and (2, 1).
Therefore, out of the total of 15 outcomes, 2 outcomes have a 3 rolled and satisfy the condition. Therefore, the probability of rolling a 3 given that the sum of the numbers is less than or equal to 6 is 2/15 or approximately 0.1333.
131
two ways to get a sum of 3 either 12 or 21
our possible outcomes, sum ≤ 6 :
11, 12, 13, 14, 15
21, 22, 23, 24
31,32, 33,
41, 42
51
15 of them
prob(your event) = 2/15