Two dice are rolled and the sum of the outcomes is counted.
(answers to 4 decimal places.)
a.)Find the probability that the sum is divisible by 3.
b.)Find the probability that the sum is smaller than 4.
c.)Twodiced rolled, min2, max12, divided by3: 3, 6, 9, 12
there are 36 possible rolls. So just check the successes
(a) there are 7 successes (12 15 21 24 33 42 51) so P = 7/36
Now you try the others
To find the probability of the sum of two dice outcomes, we need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario. Let's break down each question:
a.) Find the probability that the sum is divisible by 3.
To answer this, we need to determine the number of favorable outcomes (sums divisible by 3) and divide it by the total number of outcomes.
Total number of outcomes when two dice are rolled: 6 * 6 = 36 (as each die has 6 faces)
Favorable outcomes for sums divisible by 3:
We can approach this by listing all possible sums: (1, 1), (1, 2), (1, 3), ..., (6, 5), (6, 6)
Out of these 36 possible outcomes, we have 12 favorable sums: (1, 2, 3, 4, 5, 6), (2, 1, 3, 4, 5, 6), (3, 1, 2, 4, 5, 6), (4, 1, 2, 3, 5, 6), (5, 1, 2, 3, 4, 6), (6, 1, 2, 3, 4, 5)
So, the probability of getting a sum divisible by 3 is: 12/36 = 1/3 = 0.3333 (approximated to four decimal places).
b.) Find the probability that the sum is smaller than 4.
Again, we need to determine the favorable outcomes (sums smaller than 4) and divide it by the total outcomes.
Favorable outcomes for sums smaller than 4: (1, 1), (1, 2), (2, 1)
So, the probability of getting a sum smaller than 4 is: 3/36 = 1/12 = 0.0833 (approximated to four decimal places).
c.) Find the probability for each sum (2, 3, 4, ..., 12) divided by 3 (2, 3, 4)
To answer this, we need to list down the favorable outcomes (sums divisible by 3), and then divide each count by 36 (total outcomes).
For each sum (2, 3, 4, ..., 12), we can list down the favorable outcomes:
Sum = 2: The only favorable outcome is (1, 1). Probability = 1/36 = 0.0278
Sum = 3: The favorable outcomes: (1, 2), (2, 1). Probability = 2/36 = 0.0556
Sum = 4: The favorable outcomes: (1, 3), (2, 2), (3, 1). Probability = 3/36 = 0.0833
Sum = 5: The favorable outcomes: (1, 4), (2, 3), (3, 2), (4, 1). Probability = 4/36 = 0.1111
Sum = 6: The favorable outcomes: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1). Probability = 5/36 = 0.1389
Sum = 7: The favorable outcomes: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). Probability = 6/36 = 0.1667
Sum = 8: The favorable outcomes: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). Probability = 5/36 = 0.1389
Sum = 9: The favorable outcomes: (3, 6), (4, 5), (5, 4), (6, 3). Probability = 4/36 = 0.1111
Sum = 10: The favorable outcomes: (4, 6), (5, 5), (6, 4). Probability = 3/36 = 0.0833
Sum = 11: The favorable outcomes: (5, 6), (6, 5). Probability = 2/36 = 0.0556
Sum = 12: The only favorable outcome is (6, 6). Probability = 1/36 = 0.0278
Therefore, the probabilities for each sum divided by 3 are:
(2, 3, 4): 0.0278, 0.0556, 0.0833
(5, 6, 7): 0.1111, 0.1389, 0.1667
(8, 9, 10): 0.1389, 0.1111, 0.0833
(11, 12): 0.0556, 0.0278
To find the probabilities, we need to calculate the total number of favorable outcomes and divide it by the total number of possible outcomes.
a) Find the probability that the sum is divisible by 3:
To calculate the number of favorable outcomes, we need to determine the pairs of numbers on the two dice that add up to a number divisible by 3.
The possible outcomes are:
(1, 2), (1, 5), (1, 8), (2, 1), (2, 4), (2, 7), (3, 3), (3, 6), (3, 9), (4, 2), (4, 5), (5, 1), (5, 4), (5, 7), (5, 10), (6, 3), (6, 6), (6, 9), (7, 2), (7, 5), (7, 8), (8, 1), (8, 4), (9, 3), (9, 6), (9, 9), (10, 5), (10, 8), (11, 7), (11, 10), (12, 6), (12, 9)
Out of these 36 outcomes, 18 outcomes have a sum divisible by 3. So the number of favorable outcomes is 18.
Since there are 36 possible outcomes (6 options for the first dice and 6 options for the second dice), the probability that the sum is divisible by 3 is 18/36 = 0.5000.
b) Find the probability that the sum is smaller than 4:
To calculate the number of favorable outcomes, we need to identify the pairs of numbers on the two dice that add up to a number smaller than 4.
The possible outcomes are:
(1, 1), (1, 2), (2, 1), (2, 2), (1, 3), (3, 1)
Out of these 36 outcomes, 6 outcomes have a sum smaller than 4. So the number of favorable outcomes is 6.
Therefore, the probability that the sum is smaller than 4 is 6/36 = 0.1667.
c) The sum of the probabilities for rolling two dice and dividing by 3:
To calculate this, we need to find the probabilities for each sum of 3, 6, 9, and 12 and add them together.
The probabilities for each sum are:
Probability of the sum being 3: 2/36 = 0.0556
Probability of the sum being 6: 5/36 = 0.1389
Probability of the sum being 9: 4/36 = 0.1111
Probability of the sum being 12: 1/36 = 0.0278
Adding these probabilities together, we get 0.0556 + 0.1389 + 0.1111 + 0.0278 = 0.3333.