a pair of dice is rolled. what is the probabilty of each of the following? a. the sum of the numbers shown uppermost is less than 5. b. at least one six is cast.

out of 36 possible rolls, only 4 have the sum < 5:

12 13 21 22

What's the chance of no 6's?
5/6 * 5/6
so, take 1-P(~6,~6) = 11/36

To find the probabilities of each of these events, we need to determine the total number of favorable outcomes for each event and divide it by the total number of possible outcomes.

a. The sum of the numbers shown uppermost is less than 5.
To calculate this probability, we need to find the favorable outcomes. The possible sums less than 5 are 2, 3, and 4. We can find the number of favorable outcomes by examining all the possible combinations of two dice:

Favorable outcomes: (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)

Since there are 6 favorable outcomes, we can proceed to calculate the total number of possible outcomes. Since two dice are being rolled, each with 6 faces, the total number of outcomes is 6*6 = 36.

Therefore, the probability of rolling a sum less than 5 is equal to the number of favorable outcomes divided by the total number of outcomes:

P(sum less than 5) = (#favorable outcomes) / (#total outcomes) = 6 / 36 = 1 / 6 ≈ 0.167

So, the probability of rolling a sum less than 5 is approximately 0.167 or 16.7%.

b. At least one six is cast.
To calculate this probability, we need to find the favorable outcomes.
To find the number of favorable outcomes, we can examine the ways in which we can roll at least one six. We can calculate this by finding the number of outcomes where no six is rolled and subtracting it from the total number of outcomes.

The number of outcomes where no six is rolled is found by calculating the number of outcomes where each die roll is anything other than a six. Since there are 5 outcomes that are not a six (1, 2, 3, 4, 5) and two dice are being rolled, the total number of outcomes where no six is rolled is 5*5 = 25.

Therefore, the number of favorable outcomes where at least one six is rolled is given by subtracting the outcomes with no six from the total number of outcomes:

#favorable outcomes = #total outcomes - #no six outcomes = 36 - 25 = 11

So, the probability of rolling at least one six is equal to the number of favorable outcomes divided by the total number of outcomes:

P(at least one six) = (#favorable outcomes) / (#total outcomes) = 11 / 36 ≈ 0.306

Therefore, the probability of rolling at least one six is approximately 0.306 or 30.6%.