Suppose you roll a fair die 18 times.
(a) What is the sample space?
(b) What is the probability that each number from 1 to 6 appears exactly thrice?
(c) What is the probability that the number 1 appears exactly twice, 3 appears exactly four
times, and 5 appears exactly six times?
To answer these questions, we need to understand the concept of sample space and permutations.
(a) Sample Space:
The sample space consists of all possible outcomes of an experiment. In this case, rolling a fair die 18 times. Each roll can result in any number from 1 to 6, so for a single roll, the sample space is {1, 2, 3, 4, 5, 6}. Since we are rolling the die 18 times, the sample space for all 18 rolls is the Cartesian product of the individual rolls.
The number of elements in the sample space is given by 6^18 since there are 6 possible outcomes for each roll and we are rolling the die 18 times.
(b) Probability of Each Number Appearing Exactly Thrice:
To calculate this probability, we need to find the number of successful outcomes and divide it by the total number of possible outcomes.
To find the number of successful outcomes, we need to determine how many ways we can arrange the numbers 1 to 6 into groups of three, so that each number appears exactly three times.
We can calculate this using permutations:
Number of ways to arrange numbers 1 to 6 into groups of three = (6 choose 1) * (5 choose 1) * (4 choose 1) * (3 choose 1) * (2 choose 1) * (1 choose 1) = 6!
Given that there are 6! successful outcomes and 6^18 total outcomes, the probability is:
Probability = 6! / 6^18
(c) Probability of Specific Appearances of Numbers:
To calculate this probability, we need to find the number of successful outcomes and divide it by the total number of possible outcomes.
We can approach this by considering the number of ways each number can appear in the 18 rolls:
Number of ways number 1 appears exactly twice = (18 choose 2)
Number of ways number 3 appears exactly four times = (16 choose 4)
Number of ways number 5 appears exactly six times = (12 choose 6)
Since these events are independent, we can multiply the individual probabilities to get the final probability:
Probability = (18 choose 2) * (16 choose 4) * (12 choose 6) / 6^18