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Chapter 8: Problem 427

If a pair of dice is tossed twice, find the probability of obtaining 5 on both tosses.

Short Answer

Expert verified
The probability of obtaining a sum of 5 on both tosses of a pair of dice is \(\frac{4}{324}\) or approximately 0.0123.

Step by step solution

01

Calculate the total number of possible outcomes after each pair of dice is tossed

Each die has 6 faces and since we have a pair of dice, the total possible outcomes after a single toss are 6 x 6 = 36 outcomes. As the pair of dice is tossed twice, the total number of possible outcomes after the second toss will also be 36. Therefore, after the two tosses, there are a total of 36 x 36 = 1296 possible outcomes.
02

Determine the outcomes where the sum is 5

Now, let's find out the number of ways we can get a sum of 5 after each toss. We want to find all the combinations of numbers (x, y) such that x + y = 5, where x and y are the numbers shown on the two dice. The possible combinations are as follows: - Die 1 shows 1 and Die 2 shows 4 - Die 1 shows 2 and Die 2 shows 3 - Die 1 shows 3 and Die 2 shows 2 - Die 1 shows 4 and Die 2 shows 1 So there are 4 ways to get a sum of 5 when a pair of dice is tossed once.
03

Calculate the number of favorable outcomes for both tosses

Since the two tosses are independent of each other, the number of favorable outcomes for both tosses is simply the product of the favorable outcomes for each toss. In this case, it is 4 (from the first toss) x 4 (from the second toss) = 16 favorable outcomes.
04

Compute the probability

Now that we have both the number of favorable outcomes and the total number of possible outcomes, we can calculate the probability. The probability of obtaining a sum of 5 on both tosses is: Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 16/1296 Divide both numbers by 4 to simplify the fraction: Probability = 4/324 So, the probability of obtaining a sum of 5 on both tosses is 4/324 or approximately 0.0123 (rounded to 4 decimal places).

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