Problem 23 - A First Course in Probability

Question

A pair of fair dice is rolled. What is the probability that the second die lands on a higher value than does the first?

Step-by-Step Solution

Answer

The probability that the second die lands on a higher value than the first one when rolling a pair of fair dice is \(\frac{5}{12}\), or approximately 0.4167 (rounded to four decimal places).

Step 1: Define the sample space S

The sample space S for rolling two dice can be found by considering all possible outcomes when rolling two dice. Since there are 6 faces on each die, there are a total of 6 x 6 = 36 possible outcomes when we roll them at the same time.

Step 2: Define the event A - second die lands on a higher value than the first

Let's define event A as the second die shows a higher value than the first one. To find the probability of event A, we need to count the number of outcomes where the second die shows a higher value than the first die.

Step 3: Count the number of favorable outcomes for event A

Here, we'll list the favorable outcomes where the second die shows a higher value than the first one: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 3), (2, 4), (2, 5), (2, 6) (3, 4), (3, 5), (3, 6) (4, 5), (4, 6) (5, 6) We have a total of 15 favorable outcomes for event A.

Step 4: Calculate the probability of event A

Now that we have the total number of favorable outcomes for event A and the sample space size, we can calculate the probability of event A occurring. Recall that the probability of an event occurring is the ratio of the number of favorable outcomes to the total number of possible outcomes: \(P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{15}{36}\) To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 3: \(P(A) = \frac{15}{36} = \frac{5}{12}\)

Step 5: Interpret the result

The probability that the second die lands on a higher value than the first one when rolling a pair of fair dice is 5/12, or approximately 0.4167 (rounded to four decimal places). This means that there is approximately a 41.67% chance that the second die will show a higher value than the first one when rolling two fair dice.