The question asks about the living arrangements of single people over the age of thirty-five, based on a poll. The poll found that 34% rented a house or apartment, 21% owned a house, and 17% owned a condominium. We are then asked to consider what happens when four single people are randomly selected, with replacement.
When we select a person "with replacement," it means that after selecting a person, we put them back into the pool before making the next selection. This means that each selection is independent and does not affect the probabilities of future selections.
To answer the question, we need to calculate the probabilities of different combinations of living arrangements for the four single people.
Let's consider the first person. Since 34% rented a house or apartment, the probability of the first person renting is 0.34. Similarly, the probabilities for the first person owning a house and owning a condominium are 0.21 and 0.17, respectively.
Now, for the second person, we have the same probabilities as the first person since the selections are made with replacement. So, the probability of the second person renting is also 0.34, owning a house is 0.21, and owning a condominium is 0.17.
Following the same logic, the probabilities for the third and fourth persons are also 0.34, 0.21, and 0.17 for renting, owning a house, and owning a condominium, respectively.
To calculate the probabilities of different combinations, we multiply the probabilities of each individual person. For example, the probability that all four people rent is 0.34 * 0.34 * 0.34 * 0.34 = 0.34^4. Similarly, the probabilities for other combinations can be calculated.
Here are the probabilities for different combinations:
1. All four people rent: 0.34^4
2. Three people rent and one person owns a house: 0.34^3 * 0.21
3. Two people rent, one person owns a house, and one person owns a condominium: 0.34^2 * 0.21 * 0.17
4. Two people rent and two people own a condominium: 0.34^2 * 0.17^2
5. One person rents, two people own a house, and one person owns a condominium: 0.34 * 0.21^2 * 0.17
6. One person rents, one person owns a house, and two people own a condominium: 0.34 * 0.21 * 0.17^2
7. One person rents, three people own a condominium: 0.34 * 0.17^3
8. Two people own a house and two people own a condominium: 0.21^2 * 0.17^2
These are the probabilities for the different combinations of living arrangements for the four single people selected randomly with replacement.
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