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 5
and
(b)at least one 5 is cast
To find the probabilities for the given events, we need to determine the number of favorable outcomes and the total number of possible outcomes.
(a) The sum of the numbers shown uppermost is less than 5.
To find the favorable outcomes, we need to list all the combinations of numbers that add up to a sum less than 5. These combinations are (1, 1), (1, 2), (2, 1), (1, 3), (3, 1), (2, 2), and (1, 4), (4, 1), giving us a total of 8 favorable outcomes.
To find the total number of possible outcomes, we multiply the number of outcomes for each die. Since each die has 6 possible outcomes, the total number of outcomes is 6 x 6 = 36.
Therefore, the probability of the sum of the numbers shown uppermost being less than 5 is 8/36.
P(a) = 8/36 ≈ 0.222
(b) At least one 5 is cast.
To find the favorable outcomes, we need to find all the combinations where at least one 5 is rolled. This could be (5, 1), (5, 2), (5, 3), (5, 4), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (5, 6), (6, 5). There are 11 favorable outcomes.
Again, the total number of possible outcomes is 6 x 6 = 36.
Therefore, the probability of at least one 5 being cast is 11/36.
P(b) = 11/36 ≈ 0.306
Remember to always simplify fractions and round the probability to the desired decimal places.