Answer:
Explanation:
a) The probability that all 5 dice are the same can be calculated by considering the number of favorable outcomes divided by the total number of possible outcomes.
Each die has 6 possible outcomes (numbers 1 to 6). To have all 5 dice showing the same number, there is only 1 favorable outcome (e.g., all dice showing 1, or all dice showing 2, and so on).
Therefore, the probability that all 5 dice are the same is 1 out of 6^5, which can be simplified to 1/6^5 or approximately 0.0001286.
b) The probability that at least 4 dice are the same can be calculated by considering the number of favorable outcomes divided by the total number of possible outcomes.
To find the favorable outcomes, we need to consider two cases:
1) All 5 dice showing the same number (as calculated in part a)
2) Four dice showing the same number and the fifth die showing a different number.
For the first case, we have already determined that the probability is 1/6^5.
For the second case, we have 6 choices for the number that appears on the four dice (e.g., 1, 2, 3, 4, 5, or 6). The fifth die can show any of the remaining 5 numbers. Therefore, there are 6*5 = 30 favorable outcomes for this case.
Adding the favorable outcomes from both cases, we get a total of 31 favorable outcomes.
The total number of possible outcomes is still 6^5.
Therefore, the probability that at least 4 dice are the same is 31 out of 6^5, which can be simplified to 31/6^5 or approximately 0.0001609.
c) To calculate the probability of having exactly 4 contiguous numbers, we need to consider the number of favorable outcomes divided by the total number of possible outcomes.
For 4 numbers to be contiguous, we can start with any number from 1 to 3 (as we are considering 4 numbers) and the remaining number can be any of the next 3 numbers.
Therefore, there are 3 choices for the starting number and 3 choices for the remaining number.
Hence, the total number of favorable outcomes is 3 * 3 = 9.
The total number of possible outcomes is still 6^5.
Therefore, the probability of having exactly 4 contiguous numbers is 9 out of 6^5, which can be simplified to 9/6^5 or approximately 0.0003863.