Question

Suppose that adam, bernie, charles, and dan have annual allowances of $1000, $500, $200, and $1300, respectively. If two of these young men are chosen at random to be roommates in room 101A, and if X is the average of their annual allowances, then what is the probability function of X?

Options:
A) P(X = $250)
B) P(X = $550)
C) P(X = $650)
D) P(X = $800)

Answer 1

The probability function of X yields probabilities for specific averages of allowances between pairs of people: Adam, Bernie, Charles, and Dan. Since none of the roommate combinations result in averages of $250, $550, $650, or $800, the probabilities for these averages are zero.

Step-by-step explanation:

The question asks for the probability function of the variable X, which represents the average of the annual allowances of two people chosen at random from a group of four: Adam ($1000), Bernie ($500), Charles ($200), and Dan ($1300). To find this, we first calculate the possible averages of the annual allowances for all combinations of two people. Then, we can list the probabilities corresponding to each possible average.

We have the following possible combinations:

Adam & Bernie: Average = $(1000+500)/2 = $750

Adam & Charles: Average = $(1000+200)/2 = $600

Adam & Dan: Average = $(1000+1300)/2 = $1150

Bernie & Charles: Average = $(500+200)/2 = $350

Bernie & Dan: Average = $(500+1300)/2 = $900

Charles & Dan: Average = $(200+1300)/2 = $750

Next, determine the probabilities for each possible average. There are a total of 6 combinations (or events), so each event has a probability of 1/6.

Using this, we can answer the question. The probability function of X can only take on the values calculated. For example, P(X = $250) is zero because there are no combinations of roommates that result in an average of $250. Continuity, P(X = $550), P(X = $650), and P(X = $800) are all zero for the same reason.

Answer 2

The probability function of X yields probabilities for specific averages of allowances between pairs of people: Adam, Bernie, Charles, and Dan. Since none of the roommate combinations result in averages of $250, $550, $650, or $800, the probabilities for these averages are zero.

Step-by-step explanation:

The question asks for the probability function of the variable X, which represents the average of the annual allowances of two people chosen at random from a group of four: Adam ($1000), Bernie ($500), Charles ($200), and Dan ($1300). To find this, we first calculate the possible averages of the annual allowances for all combinations of two people. Then, we can list the probabilities corresponding to each possible average.

We have the following possible combinations:

Adam & Bernie: Average = $(1000+500)/2 = $750

Adam & Charles: Average = $(1000+200)/2 = $600

Adam & Dan: Average = $(1000+1300)/2 = $1150

Bernie & Charles: Average = $(500+200)/2 = $350

Bernie & Dan: Average = $(500+1300)/2 = $900

Charles & Dan: Average = $(200+1300)/2 = $750

Next, determine the probabilities for each possible average. There are a total of 6 combinations (or events), so each event has a probability of 1/6.

Using this, we can answer the question. The probability function of X can only take on the values calculated. For example, P(X = $250) is zero because there are no combinations of roommates that result in an average of $250. Continuity, P(X = $550), P(X = $650), and P(X = $800) are all zero for the same reason.