can you explain the following in a simpler way:
To sample values A, B, C, D based on their assigned probabilities, you need to ensure that each value is drawn with the appropriate frequency over many trials. Given that random number generation typically produces uniform results over a continuous range (e.g., between 0 and 1), you’ll need a way to map these uniformly generated values to your non-uniformly distributed values A, B, C, D.
This is where cumulative probabilities come into play. By converting individual probabilities into cumulative probabilities, we create "bins" or "intervals" on the probability range [0, 1) that correspond to each value.
Imagine you have a bag with different colored balls: A, B, C, and D. Each color ball has a different chance of being picked when you reach into the bag without looking. What you want is to pick out a ball and have it be A, B, C, or D with certain chances each time you pick.
Let's say the chance for each ball to be picked is as follows:
- A: 10% chance (or probability 0.10)
- B: 20% chance (or probability 0.20)
- C: 30% chance (or probability 0.30)
- D: 40% chance (or probability 0.40)
Now, let's think about using a random number to pick the ball. If you have a way to get random numbers between 0 and 1, like rolling a special dice that has numbers from 0 to 1 instead of 1 to 6, you can use these numbers to decide which ball to pick, kind of like where the dice lands tells you what color ball to grab.
But since the dice rolls numbers evenly from 0 to 1 (like it's just as likely to land on 0.3 as 0.7), you have to figure out a way to make sure that the uneven chances for each ball (10% for A, etc.) work with the even dice rolls.
To do this, you create "bins" using the "cumulative probability." This means you add up the chances for the balls as you go along:
- For A, your bin is 0 to 0.10, because it's just a 10% chance.
- For B, you add its 20% chance to A's, so B's bin is 0.10 to 0.30 (the previous 10% plus B's 20%).
- For C, do it again: C's bin is 0.30 to 0.60 (the previous 30% plus C's 30%).
- Finally, D's bin will be 0.60 to 1.00 (the previous 60% plus D's 40%).
Now, when you roll your special dice and get a number:
- If it's between 0 to 0.10, you pick A.
- If it's from 0.10 to 0.30, you pick B.
- If it's from 0.30 to 0.60, you pick C.
- If it's from 0.60 to 1.00, you pick D.
This is how you use the even dice rolls to make sure the balls are picked according to their different chances. The "bins" or "intervals" you created translate the random numbers into choices that match the balls' probabilities.