Find the probability a random apartment has cable or is on the second floor. There are 112 total apartments.
| | 1st | 2nd | 3rd |
| :---- | :-- | :-- | :-- |
| Cable | 16 | 13 | 14 |
| No Cable | 21 | 26 | 22 |
[tex]P(\text { Cable or 2nd Floor })=[/tex]?
Answer (2)
Calculate the probability of an apartment having cable: 112 43 ≈ 0.384 .
Calculate the probability of an apartment being on the second floor: 112 39 ≈ 0.348 .
Calculate the probability of an apartment having cable and being on the second floor: 112 13 ≈ 0.116 .
Use the formula P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) to find the probability of an apartment having cable or being on the second floor: 112 43 + 112 39 − 112 13 = 0.616 .
Explanation
Understand the problem We are given a table showing the distribution of apartments with and without cable on the 1st, 2nd, and 3rd floors. We need to find the probability that a randomly selected apartment has cable or is on the second floor.
Define events and the formula Let A be the event that an apartment has cable, and B be the event that an apartment is on the second floor. We want to find P ( A ∪ B ) , which is the probability that an apartment has cable or is on the second floor. We can use the formula: P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) where: P ( A ) is the probability that an apartment has cable. P ( B ) is the probability that an apartment is on the second floor. P ( A ∩ B ) is the probability that an apartment has cable and is on the second floor.
Calculate P(A) First, let's find the total number of apartments with cable. From the table, we have: Cable on 1st floor: 16 Cable on 2nd floor: 13 Cable on 3rd floor: 14 So, the total number of apartments with cable is 16 + 13 + 14 = 43 . Therefore, the probability that an apartment has cable is: P ( A ) = Total number of apartments Total number of apartments with cable = 112 43 ≈ 0.3839
Calculate P(B) Next, let's find the total number of apartments on the second floor. From the table, we have: Cable on 2nd floor: 13 No cable on 2nd floor: 26 So, the total number of apartments on the second floor is 13 + 26 = 39 . Therefore, the probability that an apartment is on the second floor is: P ( B ) = Total number of apartments Total number of apartments on the second floor = 112 39 ≈ 0.3482
Calculate P(A and B) Now, let's find the number of apartments that have cable and are on the second floor. From the table, we have 13 apartments with cable on the second floor. Therefore, the probability that an apartment has cable and is on the second floor is: P ( A ∩ B ) = Total number of apartments Number of apartments with cable on the second floor = 112 13 ≈ 0.1161
Calculate P(A or B) and conclude Now we can substitute these probabilities into the formula for P ( A ∪ B ) :
P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) = 112 43 + 112 39 − 112 13 = 112 43 + 39 − 13 = 112 69 ≈ 0.6161 Rounding to the nearest thousandth, we get 0.616.
Examples
In urban planning, understanding the probability of residents having specific amenities like cable TV or living on a particular floor can help in resource allocation and service optimization. For instance, if a building management wants to upgrade internet services, knowing the likelihood of residents having cable can inform decisions about infrastructure investments and marketing strategies. Similarly, understanding floor-wise distribution can aid in planning elevator maintenance or emergency evacuation procedures. These probabilities provide valuable insights for efficient management and resource distribution within residential buildings.
The probability that a random apartment has cable or is on the second floor is approximately 0.616. This is calculated by finding the probabilities of each event and using the union formula. Thus, about 61.6% of apartments fit this criterion.
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