Final answer:
To calculate the probability function of X for the average allowance of randomly chosen roommates, we examine all unique pairs of allowances, calculate their averages, and determine the likelihood of each average happening.
Step-by-step explanation:
The question presents a scenario with four individuals, each with a different annual allowance. To determine the probability function of X, we need to find the average of the annual allowances for every possible pairing of these individuals. There are six unique pairings as follows: Adam & Bernie, Adam & Charles, Adam & Dan, Bernie & Charles, Bernie & Dan, and Charles & Dan. The averages for these pairs are: ($1000 + $500)/2
= $750, ($1000 + $200)/2
= $600, ($1000 + $1300)/2
= $1150, ($500 + $200)/2
= $350, ($500 + $1300)/2
= $900, and ($200 + $1300)/2
= $750.
Because each pairing is equally likely, we can now calculate the probability for each possible value of X. The probability for an average of $350, $600, $750, $900, or $1150 would be 1/6, since there is one such occurrence out of six possible pairings. However, $750 can occur in two different pairs (Adam & Bernie, Charles & Dan), so its probability would be 2/6 or 1/3. The probability distribution of X, therefore, represents the likelihood of each possible average allowance when two individuals are randomly chosen to be roommates.