Let the random variable X be the number of rooms in a randomly chosen owner-occupied housing unit in a certain city. The distribution for the units is given below.
X 3 4 5 6 7 8 9 10
P(X) 0.07 0.23 0.36 0.19 0.05 0.05 0.03 ?
(a) Is X a discrete or continuous random variable?
(b) What must be the probability of choosing a unit with 10 rooms?
(c) What is the probability that a unit chosen at random is not a 10-room unit?
(d) What is the probability that a unit chosen at random has less than five rooms?
(e) What is the probability that a unit chosen at random has three rooms?

Probabilities are used to determine the chances of an event.

(a) Is X a discrete or a continuous random variable

From the table, the values of  X are natural numbers.

Only discrete random variables take natural numbers for their dataset.

Hence, X is a discrete random variable

(b) P(x = 10)

From the table, we have:

Using the complement rule, we have:

So, we have:

Hence, the probability of choosing a unit with 10 rooms must be 0.08

(c) P (x ! = 10)

In (b), we have

This means that:

The probability of choosing a unit that is not 10 rooms is 0.92

(d) P(x < 5)

This is calculated as:

So, we have:

Hence, the probability of choosing a unit that has less than 5 rooms is 0.30

(e) P(x = 3)

From the table, we have:

Hence, the probability of choosing a unit that has 3 rooms is 0.07

Read more about probabilities at:

link

a) X is a discrete random variable.

b) 2% probability of choosing a unit with 10 rooms

c) 98% probability that a unit chosen at random is not a 10-room unit

d) 30% probability that a unit chosen at random has less than five rooms

e) 7% probability that a unit chosen at random has less than five rooms

Step-by-step explanation:

The distribution means that:

7% probability that a randomly chosen owner-occupied housing unit in a certain city has 3 rooms.

23% probability that a randomly chosen owner-occupied housing unit in a certain city has 4 rooms.

36% probability that a randomly chosen owner-occupied housing unit in a certain city has 5 rooms.

19% probability that a randomly chosen owner-occupied housing unit in a certain city has 6 rooms.

5% probability that a randomly chosen owner-occupied housing unit in a certain city has 7 rooms.

5% probability that a randomly chosen owner-occupied housing unit in a certain city has 8 rooms.

3% probability that a randomly chosen owner-occupied housing unit in a certain city has 9 rooms.

?% probability that a randomly chosen owner-occupied housing unit in a certain city has 10 rooms.

(a) Is X a discrete or continuous random variable?

X is the number of rooms in a randomly selected house. The number of rooms is a countable variable, that is, only assumes values like 1,2,3,4,... You cannot have 3.5 rooms, for example.

So X is a discrete random variable.

(b) What must be the probability of choosing a unit with 10 rooms?

The sum of all probabilities must be 100%(1 decimal). So

0.07 + 0.23 + 0.36 + 0.19 + 0.05 + 0.05 + 0.03 + ? = 1

0.98 + ? = 1

? = 0.02

2% probability of choosing a unit with 10 rooms

(c) What is the probability that a unit chosen at random is not a 10-room unit?

0.07 + 0.23 + 0.36 + 0.19 + 0.05 + 0.05 + 0.03 = 0.98

98% probability that a unit chosen at random is not a 10-room unit

(d) What is the probability that a unit chosen at random has less than five rooms?

7% probability it has 3 rooms

23% probability it has 4 rooms. So

7+23 = 30% probability that a unit chosen at random has less than five rooms

(e) What is the probability that a unit chosen at random has three rooms?

7% probability that a randomly chosen owner-occupied housing unit in a certain city has 3 rooms. So

7% probability that a unit chosen at random has less than five rooms

(a) the volume of the cone is 20/3 in3

hence, option a is correct.