Question

What is the probability of getting a sum of 22 or more when four dice are thrown?

• A. \(\frac{5}{432}\)
• B. \(\frac{4}{648}\)
• C. \(\frac{1}{144}\)
• D. \(\frac{7}{36}\)

Answer 1

The Correct Option is A

To solve the problem of finding the probability of getting a sum of 22 or more when four dice are thrown, we need to understand a few basic principles of probability: 

1. **Total Number of Outcomes**: When a single die is thrown, the possible outcomes are 1, 2, 3, 4, 5, and 6, resulting in a total of 6 outcomes. Therefore, when four dice are thrown, the total number of possible outcomes is \(6^4\).

2. **Calculating Possible Combinations for Desired Outcome**: We are looking for a sum of 22 or more. The maximum sum possible with four dice is 24 (i.e., \(6 + 6 + 6 + 6\)). Therefore, we will need to calculate the number of ways to achieve sums of 22, 23, and 24.

3. **Calculating for Each Possible Sum**:

• Sum = 24: There's only one way to get a sum of 24, which is by getting a 6 on all four dice: \((6, 6, 6, 6)\).
• Sum = 23: There are four potential combinations to obtain this sum, each combination involves three dice showing 6 and one die showing 5. The possible outcomes are \((6, 6, 6, 5)\).
• Sum = 22: Different combinations can compose 22, such as two dice showing 6 and two showing 5, with permutations among them. After accountability, there are 10 combinations.

4. **Add the Possible Combinations**: Since these sums can be achieved by different methods and are mutually exclusive (you can only have one sum per throw), we can simply add the number of possible outcomes:

Total Desired Outcomes = 1 (for 24) + 4 (for 23) + 10 (for 22) = 15.

5. **Calculate the Probability**: The probability is the ratio of the desired outcomes to the total outcomes:

\(\text{Probability} = \frac{15}{6^4} = \frac{15}{1296} = \frac{5}{432}\)

Thus, the correct probability of getting a sum of 22 or more is \(\frac{5}{432}\), confirming the correctness of the first option.