A pair of fair dice is rolled. What is the probability of each of the following? (Round your answers to three decimal places.)
(a) the sum of the numbers shown uppermost is less than 6
(b) at least one 6 is cast
There are 36 possible outcomes.
List the ones you want. For example, sum < 6:
1 4
2 3
3 2
4 1
There are 4 ways to get that, so P(sum<6) = 4/36
Same for the other one.
Well, let's take a look at these dicey situations!
(a) To find the probability that the sum of the numbers shown uppermost is less than 6, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes.
The favorable outcomes are when we have (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), and (4,1). That's a total of 10 outcomes.
For two fair dice, each with six faces, the total number of possible outcomes is 6 * 6 = 36.
So, the probability of the sum being less than 6 is 10/36 ≈ 0.278.
(b) Now, to find the probability of at least one 6 being cast, we can consider the complementary event, which is the probability of no 6 being cast.
The probability of not rolling a 6 on a single die is 5/6. Since we have two dice, the probability of not rolling a 6 on both is (5/6) * (5/6) = 25/36.
Therefore, the probability of at least one 6 being cast is 1 - 25/36 = 11/36 ≈ 0.306.
Remember, probability can be a tricky business, but it's always important to have fun with the numbers!
To find the probabilities, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.
(a) To find the probability that the sum of the numbers shown uppermost is less than 6, we need to count the number of favorable outcomes.
Favorable outcomes: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)
Total number of possible outcomes: Since there are 6 possible outcomes for each dice roll, there are a total of 6^2 = 36 possible outcomes.
Probability = Favorable outcomes / Total outcomes
= 10 / 36
= 5 / 18
≈ 0.278
Therefore, the probability that the sum of the numbers shown uppermost is less than 6 is approximately 0.278.
(b) To find the probability of rolling at least one 6, we need to find the number of outcomes where at least one 6 is rolled.
Number of outcomes with at least one 6: (1,6), (2,6), (3,6), (4,6), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
Total number of possible outcomes: 36
Probability = Favorable outcomes / Total outcomes
= 11 / 36
≈ 0.306
Therefore, the probability of rolling at least one 6 is approximately 0.306.
To find the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes.
(a) The sum of the numbers shown uppermost is less than 6.
To find the favorable outcomes, we need to consider all the possible combinations of two dice that result in a sum less than 6: (1,1), (1,2), (1,3), (2,1), (2,2), (3,1).
So, there are 6 favorable outcomes.
To find the total number of possible outcomes, we need to calculate the total number of combinations when two dice are rolled. Each die has 6 possible outcomes, so the total number of outcomes is 6*6 = 36.
The probability of the sum of the numbers shown being less than 6 is given by:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 6/36 = 1/6 ≈ 0.167
Therefore, the probability that the sum of the numbers is less than 6 is approximately 0.167.
(b) At least one 6 is cast.
To find the favorable outcomes, we need to consider all the possible combinations where at least one 6 is rolled. There are 11 favorable outcomes: (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (1,6), (2,6), (3,6), (4,6), (5,6).
The total number of possible outcomes remains 36.
The probability of getting at least one 6 is given by:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 11/36 ≈ 0.306
Therefore, the probability of rolling at least one 6 is approximately 0.306.