I'm not sure how to go about it? I think for part a, would be 16C5, or am I wrong?
The outcome of chance events in a fantasy role-playing game is determined by rolling polyhedral dice with anywhere from 4 to 20 sides. Suppose you roll a 16 sided die 5 times and observe the number on the top of the die.
a) How many possible outcomes are there for these 5 rolls?
b) In how many of the outcomes in the part (a) do the 5 rolls produce five different numbers?
c) What is the probability that at least 2 of the rolls are the same?
To calculate the number of possible outcomes for these 5 rolls, we need to consider the number of choices for each roll. Since you are rolling a 16-sided die 5 times, each roll has 16 possible outcomes. Therefore, the total number of outcomes for the 5 rolls can be calculated by multiplying the number of outcomes for each roll:
a) Number of possible outcomes = (Number of outcomes per roll) ^ (Number of rolls)
Number of possible outcomes = 16^5 = 1048576
So, there are 1,048,576 possible outcomes for these 5 rolls.
For part b, we need to determine how many of these outcomes produce five different numbers. Since each roll has 16 possible outcomes, the first roll can be any number from 1 to 16. After that, the second roll cannot be the same as the first roll, so it has 15 possible outcomes. Similarly, the third roll has 14 possible outcomes, the fourth roll has 13 possible outcomes, and the fifth roll has 12 possible outcomes. Therefore, the number of outcomes with five different numbers can be calculated as:
b) Number of outcomes with five different numbers = 16 * 15 * 14 * 13 * 12 = 524,160
So, there are 524,160 outcomes in which the 5 rolls produce five different numbers.
Finally, for part c, we need to calculate the probability that at least 2 of the rolls are the same. The opposite of "at least 2 of the rolls are the same" is "all rolls are different." We have already calculated the number of outcomes with five different numbers in part b, which is 524,160. Therefore, the probability that at least 2 of the rolls are the same can be calculated as:
c) Probability that at least 2 rolls are the same = 1 - (Number of outcomes with all rolls different / Total number of outcomes)
Probability that at least 2 rolls are the same = 1 - (524,160 / 1,048,576)
Probability that at least 2 rolls are the same = 0.4999 (rounded to 4 decimal places)
Therefore, the probability that at least 2 of the rolls are the same is approximately 0.4999 or 49.99%.
To determine the number of possible outcomes for rolling a 16-sided die 5 times, we can use the concept of counting principles.
a) In this case, each roll can result in any number from 1 to 16, and there are 5 independent rolls. To calculate the number of possible outcomes, we need to find the total number of choices for each roll (16 in this case) and multiply them together since each roll is independent. Therefore, the answer is 16^5 or 16 * 16 * 16 * 16 * 16 = 1,048,576 possible outcomes.
So, your answer for part (a) would be 1,048,576, instead of 16C5, which represents selecting a combination of 5 items out of a set of 16.
b) To calculate the number of outcomes where the 5 rolls produce five different numbers, we need to consider the number of choices for the first roll (16 options), the number of choices for the second roll (15 options), the third (14 options), the fourth (13 options), and the fifth (12 options). Again, we multiply these numbers together since each roll is independent. Therefore, the answer is 16 * 15 * 14 * 13 * 12 = 5,040 outcomes.
c) To calculate the probability that at least 2 of the rolls are the same, we firstly need to find the total number of outcomes where all the rolls have different numbers (which we calculated in part b) and then subtract it from the total number of possible outcomes (which we calculated in part a). Therefore, the answer is (1,048,576 - 5,040) / 1,048,576 ≈ 0.9952 or approximately 99.52%.
Remember, for probability, we divide the number of desired outcomes by the total number of possible outcomes.