The problem requires us to find the probability that when two houses are picked at random, one has a floor area greater than $130 m^2$ and the other has a floor area $60 m^2$ or less, based on the provided histogram. We first need to calculate the frequencies from the histogram.

1. Problem Description

The problem requires us to find the probability that when two houses are picked at random, one has a floor area greater than $130 m^2$ and the other has a floor area $60 m^2$ or less, based on the provided histogram. We first need to calculate the frequencies from the histogram.

2. Solution Steps

Step 1: Determine the frequencies from the histogram.

The area of each bar represents the frequency. Frequency density = frequency/class width, so Frequency = Frequency density * class width.

* 40-60: Frequency density = 0.7, class width = 20, Frequency = 0.7 * 20 = 14

* 60-80: Frequency density = 0.9, class width = 20, Frequency = 0.9 * 20 = 18

* 80-100: Frequency density = 0.4, class width = 20, Frequency = 0.4 * 20 = 8

* 100-120: Frequency density = 0.45, class width = 20, Frequency = 0.45 * 20 = 9

* 120-140: Frequency density = 0.3, class width = 20, Frequency = 0.3 * 20 = 6

* 140-200: Frequency density = 0.1, class width = 60, Frequency = 0.1 * 60 = 6

Total frequency = 14 + 18 + 8 + 9 + 6 + 6 = 61

Step 2: Calculate the number of houses with a floor area greater than 130 $m^2$.

Houses with area greater than 130 $m^2$ are those in the 140-200 range, and some of the 120-140 range. We approximate by just taking houses in the 140-200 range.

Number of houses > 130 $m^2$ = 6

Step 3: Calculate the number of houses with a floor area 60 $m^2$ or less.

Houses with area 60 $m^2$ or less are those in the 40-60 range.

Number of houses <= 60 $m^2$ = 14

Step 4: Calculate the probability of selecting one house with area > 130 $m^2$ and one with area <= 60 $m^2$.

We can pick these houses in two ways: (>130, <=60) or (<=60, >130). The order matters.

Probability(>130, <=60) = (6/61) * (14/60)

Probability(<=60, >130) = (14/61) * (6/60)

The total probability is the sum of these two probabilities:

P = (6/61) * (14/60) + (14/61) * (6/60) = 2 * (6/61) * (14/60) = 2 * (84/3660) = 168/3660 = 42/915 = 14/305

Step 5: Simplify the fraction.

14/305 is already in simplest form.

3. Final Answer

14/305