Question

A pair of dice is rolled. Determine the probability of a result with:

i. one die showing a 4 and the other a 5?
ii. both dice showing the same result;
iii: at least one die showing a result of 3;
iv: either a 4 or 6 being displayed;
v: both dice showing even numbers;
vi: the sum of the values being 7.

Answer 1

The probabilities associated with different outcomes when rolling a pair of dice range from 0.0556 for one die showing a 4 and the other a 5, to 0.5556 for either a 4 or 6 being displayed. Calculations involve determining possible combinations and sometimes their complements.

Step-by-step explanation:

The student's question pertains to calculating various probabilities when rolling a pair of fair six-sided dice. Below is the step-by-step explanation for each scenario:

1. Probability of one die showing a 4 and the other a 5 is 2/36 or 0.0556, as there are two favorable outcomes (4,5) and (5,4) out of 36 possible combinations when two dice are rolled.
2. The probability of both dice showing the same result is 6/36 or 0.1667, as there are six possibilities: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6).
3. The probability of at least one die showing a result of 3 is calculated by finding the complement of both dice not showing 3, which is 1 - (5/6 * 5/6) or 0.3056.
4. The probability of either a 4 or 6 being displayed is 20/36 or 0.5556, considering all the combinations where at least one of the dice shows a 4 or a 6.
5. The probability of both dice showing even numbers is 9/36 or 0.2500, as there are nine combinations where both are even: (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6).
6. Finally, the probability of the sum of the values being 7 is 6/36 or 0.1667. The six combinations that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

For all calculated probabilities, remember to round the values to four decimal places as per instruction.