Within a certain group of four horses numbered #1 to #4 it has been observed that the frequency with which horse #1 wins is twice that of horse #2; that this one in turn wins twice as many times as #3, and that #3 wins twice as many times as #4. Find the probability that in the next race the winning horse will be #1 or #3. with procedure

Question

GPT 3.5 · Answer

Let's denote the number of times each horse wins as follows:
Horse #1: x
Horse #2: y
Horse #3: z
Horse #4: w

According to the given information:
- The frequency with which horse #1 wins is twice that of horse #2, so x = 2y.
- Horse #2 wins twice as many times as horse #3, so y = 2z.
- Horse #3 wins twice as many times as horse #4, so z = 2w.

To find the probability that horse #1 or #3 will win the next race, we need to calculate the sum of their probabilities individually.

The probability of horse #1 winning can be calculated by dividing the number of times horse #1 wins (x) by the total number of races. Similarly, the probability of horse #3 winning can be calculated by dividing the number of times horse #3 wins (z) by the total number of races.

Since the total number of wins must equal the total number of races (assuming no ties), we can write:
x + y + z + w = x + 2z + z + 2w = 4w = total number of races.

We can assume a specific number for w to simplify the calculations. Let's assume w = 1 (you can choose any other positive integer as well). Therefore:
x + y + z + w = 4w
x + 2z + z + 2w = 4w
x + 3z = 3w
x = 3w - 3z

Now, let's substitute the values of x and z in terms of w into the probability calculations:
Probability of horse #1 winning = x / total number of races = (3w - 3z) / 4w
Probability of horse #3 winning = z / total number of races = z / 4w

To calculate the total probability of horse #1 or #3 winning, we add their individual probabilities:
Total probability = (3w - 3z) / 4w + z / 4w

Simplifying the expression:
Total probability = (3w - 3z + z) / 4w
Total probability = (3w - 2z) / 4w

Now, you can substitute the specific value of w that you assumed to find the probability. For example, if you assumed w = 1, then the total probability becomes:
Total probability = (3 - 2z) / 4

Note: The answer will depend on the specific value of w you assume. If you choose a different value for w, the probabilities will change accordingly.