Borel–Kolmogorov paradox
Imagine you have a big jar of candy with different colors - red, green, and blue. You know that the jar has exactly 30 candies, and you also know that there are 10 red candies, 10 green candies, and 10 blue candies.
Now, you want to pick two candies from the jar randomly and record the colors you get each time. You can get two candies of the same color, or you can get two candies of different colors. This is where the paradox comes in.
The Borel-Kolmogorov paradox says that the probability of getting two candies of the same color is different from the probability of getting two candies of different colors, but these probabilities seem like they should be the same.
Let's look at the math:
- The probability of getting two candies of the same color (like two red candies) is easy to figure out. You have a 1/3 chance of picking a red candy first, and if you do, then you have a 9/29 chance of picking another red candy second (because there are only 9 red candies left out of 29 total candies). So, the probability of getting two red candies is (1/3) x (9/29) = 3/29.
- The probability of getting two candies of different colors is a little bit trickier. You have a 1/3 chance of picking a red candy first, and if you do, then you have a 20/29 chance of picking a non-red candy second (because there are 20 non-red candies left out of 29 total candies). Similarly, if you pick a non-red candy first, then you have a 10/29 chance of picking a matching color candy second. So, the probability of getting two candies of different colors is (1/3) x (20/29) + (2/3) x (10/29) = 20/29.
So, the probability of getting two candies of the same color is 3/29, and the probability of getting two candies of different colors is 20/29. These probabilities are different, even though it seems like they should be the same!
This paradox happens because the two events (getting two candies of the same color and getting two candies of different colors) are not actually independent - the second event depends on what happened in the first event. This means that the probabilities aren't as simple as we might have originally thought.
So, even though it might seem surprising, the Borel-Kolmogorov paradox reminds us that we always need to be careful when calculating probabilities and taking into account all the possible outcomes.
Now, you want to pick two candies from the jar randomly and record the colors you get each time. You can get two candies of the same color, or you can get two candies of different colors. This is where the paradox comes in.
The Borel-Kolmogorov paradox says that the probability of getting two candies of the same color is different from the probability of getting two candies of different colors, but these probabilities seem like they should be the same.
Let's look at the math:
- The probability of getting two candies of the same color (like two red candies) is easy to figure out. You have a 1/3 chance of picking a red candy first, and if you do, then you have a 9/29 chance of picking another red candy second (because there are only 9 red candies left out of 29 total candies). So, the probability of getting two red candies is (1/3) x (9/29) = 3/29.
- The probability of getting two candies of different colors is a little bit trickier. You have a 1/3 chance of picking a red candy first, and if you do, then you have a 20/29 chance of picking a non-red candy second (because there are 20 non-red candies left out of 29 total candies). Similarly, if you pick a non-red candy first, then you have a 10/29 chance of picking a matching color candy second. So, the probability of getting two candies of different colors is (1/3) x (20/29) + (2/3) x (10/29) = 20/29.
So, the probability of getting two candies of the same color is 3/29, and the probability of getting two candies of different colors is 20/29. These probabilities are different, even though it seems like they should be the same!
This paradox happens because the two events (getting two candies of the same color and getting two candies of different colors) are not actually independent - the second event depends on what happened in the first event. This means that the probabilities aren't as simple as we might have originally thought.
So, even though it might seem surprising, the Borel-Kolmogorov paradox reminds us that we always need to be careful when calculating probabilities and taking into account all the possible outcomes.