Question

the table below shows the ages of houses to the nearest year in a neighborhood. using the age of the houses as the random variable, x, which graph shows the probability distribution, ( p_x(x) ), of a randomly chosen house?

age of house	number of houses
2	20
3	25
4	30

probability distribution graph (partial shown with bars for x=1,2,3,4 and y-axis from 0 to 0.35)

Explanation:

Step1: Calculate total number of houses

First, find the total number of houses by summing the number of houses for each age. So, \(15 + 20 + 25 + 30 = 90\) total houses.

Step2: Calculate probabilities for each age

• For age 1: Probability \(P(X = 1)=\frac{15}{90}\approx0.1667\)
• For age 2: Probability \(P(X = 2)=\frac{20}{90}\approx0.2222\)
• For age 3: Probability \(P(X = 3)=\frac{25}{90}\approx0.2778\)
• For age 4: Probability \(P(X = 4)=\frac{30}{90}\approx0.3333\)

Step3: Match with the graph

Check the bar heights in the given graph. The first bar (age 1) is around 0.15 - 0.16, the second (age 2) around 0.2 - 0.22, third (age 3) around 0.25 - 0.28, fourth (age 4) around 0.3 - 0.33. This matches the calculated probabilities. So the provided graph (the one shown) is the correct probability distribution graph.

Answer:

The graph shown (labeled "Probability Distribution" with bars for ages 1 - 4 having heights approximately matching the calculated probabilities \(\frac{15}{90}\), \(\frac{20}{90}\), \(\frac{25}{90}\), \(\frac{30}{90}\)) is the correct probability distribution graph for \(P_X(x)\).