Choose an American household at random, and let X be the number of persons living in the household. If we ignore the few households with more than seven inhabitants, the probability model for X is as follows: Household size X Probability 1 0.27 2 0.33 3 0.16 4 0.14 5 0.06 6 0.03 7 0.01 What is P(X < 5)? Group of answer choices
The Correct Answer and Explanation is:
To calculate P(X<5)P(X < 5)P(X<5), we add the probabilities of all household sizes less than 5. These are the probabilities for X=1,2,3,X = 1, 2, 3,X=1,2,3, and 444.P(X<5)=P(X=1)+P(X=2)+P(X=3)+P(X=4)P(X < 5) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)P(X<5)=P(X=1)+P(X=2)+P(X=3)+P(X=4)P(X<5)=0.27+0.33+0.16+0.14P(X < 5) = 0.27 + 0.33 + 0.16 + 0.14P(X<5)=0.27+0.33+0.16+0.14P(X<5)=0.90P(X < 5) = 0.90P(X<5)=0.90
Correct Answer: 0.90
Explanation:
In probability, when we are asked to find P(X<5)P(X < 5)P(X<5), it means we want the total probability that a randomly chosen household has fewer than 5 people. That is, the household could have 1, 2, 3, or 4 persons.
Each of these possible household sizes has a corresponding probability in the model. Since the household sizes are mutually exclusive (a household can only have one specific size at a time), the total probability of X<5X < 5X<5 is simply the sum of the individual probabilities for X=1X = 1X=1 through X=4X = 4X=4.
This is based on the Addition Rule of Probability for mutually exclusive events. When none of the events can happen at the same time, we can just add their probabilities to find the probability of any one of them occurring.
Here are the steps again:
- Identify all values of XXX that are less than 5.
- Look up their corresponding probabilities in the provided table.
- Add those probabilities together.
This gives us:0.27 (X=1)+0.33 (X=2)+0.16 (X=3)+0.14 (X=4)=0.900.27\ (X = 1) + 0.33\ (X = 2) + 0.16\ (X = 3) + 0.14\ (X = 4) = 0.900.27 (X=1)+0.33 (X=2)+0.16 (X=3)+0.14 (X=4)=0.90
This result means that there is a 90 percent chance that a randomly selected American household will have fewer than 5 people. This reflects the demographic trend toward smaller household sizes in the United States, with one and two-person households making up the majority.
