3 dice are rolled let (y,r,b) represent each outcome?
a- sample space
b- prob rolling a sum of 18
c- prob of rolling a sum less than 5
To answer these questions, we need to understand how many possible outcomes there are and how many of those outcomes satisfy the given conditions.
a) The sample space is the set of all possible outcomes when rolling 3 dice. Each die has 6 faces numbered 1 to 6. Since there are 3 dice, the total number of outcomes is 6 multiplied by 6 multiplied by 6, which equals 216. Therefore, the sample space has 216 possible outcomes.
b) To find the probability of rolling a sum of 18, we need to determine the number of outcomes that satisfy this condition and divide it by the total number of outcomes in the sample space.
To count the number of outcomes for a sum of 18, we need to consider all possible combinations of numbers on the dice that sum up to 18. Since the dice have 6 faces, the maximum sum we can achieve is 6 * 3 = 18. Therefore, we can have combinations like (6, 6, 6) and permutations like (6, 6, 6) or (6, 6, 6).
In this case, there is only one possible outcome that satisfies the condition, which is (6, 6, 6). Therefore, the number of outcomes that result in a sum of 18 is 1.
Now, we divide this number by the total number of outcomes in the sample space to calculate the probability:
Probability = Number of outcomes satisfying the condition / Total number of outcomes in the sample space
Probability = 1 / 216
Probability = 1/216 ≈ 0.0046 or 0.46%
Therefore, the probability of rolling a sum of 18 is approximately 0.0046 or 0.46%.
c) To find the probability of rolling a sum less than 5, we need to determine the number of outcomes that satisfy this condition and divide it by the total number of outcomes in the sample space.
As before, we need to consider all possible combinations of numbers on the dice that sum up to less than 5. The maximum sum of the three dice we can have is 6 * 3 = 18, so we need to find all outcomes that sum up to less than 5. These outcomes can be (1, 1, 1), (1, 1, 2), (1, 1, 3), (1, 2, 1), (1, 2, 2), (1, 3, 1), and so on.
By listing out all these outcomes, we can count them and determine that there are 10 outcomes that sum up to less than 5.
Now, we divide this number by the total number of outcomes in the sample space to calculate the probability:
Probability = Number of outcomes satisfying the condition / Total number of outcomes in the sample space
Probability = 10 / 216
Probability ≈ 0.0463 or 4.63%
Therefore, the probability of rolling a sum less than 5 is approximately 0.0463 or 4.63%.