A planned community has 300 homes, each with an automatic garage door opener. The door opener has eight switches that a homeowner can set to 0 or 1. For example, a door opener code might be 01100101.
Assuming all the homes in the community are sold, what is the probability that at least two homeowners will set their switches to the same code, and will therefore be able to open each other's garage doors?
100%
To find the probability that at least two homeowners will have the same code, we can use the concept of complement probability. We will calculate the probability that all homeowners have unique codes and subtract it from 1 to get the probability of at least two homeowners having the same code.
First, let's determine the number of possible combinations for the eight switches. Since each switch can be set to either 0 or 1, there are 2 possibilities for each switch. Therefore, the total number of possible combinations is 2^8 = 256.
Now, let's calculate the probability that all homeowners have unique codes. For the first homeowner, any code can be selected, so the probability is 1. For the second homeowner, there are 255 remaining codes that are unique, so the probability is 255/256. Similarly, for the third homeowner, there are 254 remaining unique codes, so the probability is 254/256. Continuing this pattern, for the 300th homeowner, there are 257 unique codes left, so the probability becomes 257/256.
To calculate the probability of all homeowners having unique codes, we can multiply these probabilities together:
Probability of all homeowners having unique codes = (1) * (255/256) * (254/256) * ... * (257/256).
Now, let's calculate this probability:
Probability of all homeowners having unique codes = (1) * (255/256) * (254/256) * ... * (257/256)
= 0.8133 (approximately).
Finally, to find the probability of at least two homeowners having the same code, we subtract the probability of all homeowners having unique codes from 1:
Probability of at least two homeowners having the same code = 1 - 0.8133
= 0.1867 (approximately).
Therefore, the probability that at least two homeowners will set their switches to the same code is approximately 0.1867, or 18.67%.
To find the probability that at least two homeowners will set their switches to the same code, we can use the concept of complementary probability. We can calculate the probability that no two homeowners will set their switches to the same code and subtract it from 1.
Let's break down the problem step by step:
Step 1: Find the total number of possible combinations for the switches on a single door opener.
There are 8 switches, and each switch can be set to 0 or 1. So, there are 2^8 = 256 possible combinations for a single door opener code.
Step 2: Find the number of ways homeowners can set their switches so that no two codes are the same.
The first homeowner can set their switches to any of the 256 possible combinations.
The second homeowner, to avoid having the same code as the first homeowner, can choose from 255 remaining combinations.
Similarly, the third homeowner can choose from 254 remaining combinations, and so on.
Therefore, the number of ways homeowners can set their switches without any two codes being the same is: 256 * 255 * 254 * ... * (256 - 299)
Step 3: Calculate the probability of no two homeowners having the same code.
The probability of no two homeowners having the same code is the number of ways they can set their switches without repetition divided by the total possible combinations.
So, the probability is: (256 * 255 * 254 * ... * (256 - 299)) / (256^300)
Step 4: Calculate the probability of at least two homeowners having the same code.
The probability of at least two homeowners having the same code is 1 minus the probability of no two homeowners having the same code.
So, the probability is: 1 - (256 * 255 * 254 * ... * (256 - 299)) / (256^300)
This probability can be quite challenging to calculate manually due to the large numbers involved. Therefore, it's recommended to use software or programming tools like Python or Excel to calculate the probability.