four fair six-sided dice are rolled. the probability that the sum of t...
Solution:
To find the probability that the sum of the results of four fair six-sided dice is 22, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.
Step 1: Determining the Total Number of Possible Outcomes
Each dice has six possible outcomes, ranging from 1 to 6. Since there are four dice, the total number of possible outcomes is 6^4 = 1296.
Step 2: Determining the Number of Favorable Outcomes
To have a sum of 22, we need to find combinations of numbers on the four dice that add up to 22.
Sub-Step 2.1: Determining the Maximum Value on a Dice
To determine the combinations, we need to find the maximum value that can be rolled on a dice. Since we have four dice, if all four dice show their maximum value, the sum would be 6 * 4 = 24. Therefore, the maximum value on a dice is 6.
Sub-Step 2.2: Enumerating the Combinations
We can enumerate the combinations in the following way:
- Assume the first dice rolls a 1. We need to find combinations on the remaining three dice that add up to 21.
- Assume the first dice rolls a 2. We need to find combinations on the remaining three dice that add up to 20.
- ...
- Assume the first dice rolls a 6. We need to find combinations on the remaining three dice that add up to 16.
Sub-Step 2.3: Using Recursion to Find Combinations
To find the combinations, we can use recursion. We start with the first dice and recursively find combinations on the remaining dice that add up to the remaining sum. The base case is when we have only one dice remaining, and the sum should be between 1 and 6 (inclusive). In this case, we have only one favorable outcome.
Step 3: Calculating the Probability
The number of favorable outcomes is the sum of the combinations obtained in Step 2. Therefore, the probability is x/1296.
Step 4: Evaluating x
To find the value of x, we need to calculate the number of favorable outcomes. Summing up all the combinations obtained in Step 2, we get x = 10.
Answer: The value of x is 10.
four fair six-sided dice are rolled. the probability that the sum of t...
Here, we have take two 6 and two 5.
the addition of those =6+6+5+5 =22.
therefore we can obtain... there is four fair and two 6 and 5 are take respectively
=4! /(2! *2!) =6 ways (2! is remove in both cases where 6 are swap themselves same as both 5 also)...
hence =6+4 =10
To make sure you are not studying endlessly, EduRev has designed Computer Science Engineering (CSE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Computer Science Engineering (CSE).