. Three numbers are chosen at random from the set {1, 2, . . . , 50}. Find the probability
that the chosen numbers are in geometric progression.
To find the probability that the chosen numbers are in geometric progression, we need to determine the total number of possible choices and the number of favorable choices.
Total Number of Possible Choices:
The total number of possible choices can be found by calculating the number of ways to select 3 numbers from the set {1, 2, ..., 50}. This can be calculated using combinations or binomial coefficients. The formula for combinations is given by C(n, r) = n! / (r!(n-r)!), where n is the number of elements to choose from and r is the number of elements to choose. In this case, n = 50 and r = 3.
Total Number of Possible Choices = C(50, 3) = 50! / (3!(50-3)!) = 19600
Number of Favorable Choices:
To find the number of favorable choices, we need to determine how many geometric progressions of 3 numbers can be formed from the set {1, 2, ..., 50}.
In a geometric progression, the ratio between consecutive terms remains constant. The ratio can be any positive integer greater than 1 and less than or equal to 50.
For a given ratio r, the first term of the geometric progression can be any number from 1 to 50. The second term can be determined by multiplying the first term by r, and the third term can be determined by multiplying the second term by r.
There are 50 possible choices for the first term, and for each choice of the first term, there are 50 / r choices for the second term (since the second term must be divisible by the ratio r). Once the first two terms are chosen, the third term is uniquely determined.
Therefore, the number of favorable choices is the sum of the number of choices for each possible ratio r.
Number of Favorable Choices = Σ [50 / r] (summed for r = 2 to 50).
Now, using this information, we can calculate the probability:
Probability = Number of Favorable Choices / Total Number of Possible Choices
Let's calculate the number of favorable choices:
Number of Favorable Choices = Σ [50 / r] (summed for r = 2 to 50)
= (50 / 2) + (50 / 3) + (50 / 4) + ... + (50 / 50)
= 25 + 16.67 + 12.5 + ... + 1
= 630
Now, we can calculate the probability:
Probability = Number of Favorable Choices / Total Number of Possible Choices
Probability = 630 / 19600
Probability ≈ 0.0321
Therefore, the probability that the chosen numbers are in geometric progression is approximately 0.0321 or 3.21%.
To find the probability that the chosen numbers are in geometric progression, we need to determine the total number of possible outcomes and the number of favorable outcomes.
We have a set of numbers from 1 to 50, and we need to select three numbers. The total number of ways to choose three numbers from the set is given by the binomial coefficient (also known as "n choose k") formula:
C(n, k) = n! / (k!(n-k)!),
where n is the total number of elements in the set and k is the number of elements we are selecting.
In this case, n = 50 and k = 3. So the total number of possible outcomes is:
C(50, 3) = 50! / (3!(50-3)!) = 19600.
Now, let's determine the number of favorable outcomes, i.e., the number of ways we can choose three numbers that form a geometric progression.
For a geometric progression, the common ratio must be non-zero. There are two possibilities for the common ratio: positive or negative.
1. Positive common ratio:
We need to find the number of ways to choose three positive numbers such that they form a geometric progression. We can count the number of possible common ratios and then determine the number of ways to choose the first term.
- The common ratio can be any number from (1, 2, 3, ..., 25), since we are considering positive numbers.
- For each common ratio, the first term can be chosen in (50/ratio) ways, as it needs to be a divisor of the ratio and lie in the range of (1, 2, 3, ..., 50).
So, the total number of favorable outcomes for positive common ratios is:
Σ(50/ratio) for ratio = 1 to 25.
2. Negative common ratio:
Similarly, we need to find the number of ways to choose three negative numbers such that they form a geometric progression. Again, we count the number of possible common ratios and the number of ways to choose the first term.
- The common ratio can be any number from (-1, -2, -3, ..., -25), since we are considering negative numbers.
- For each common ratio, the first term can be chosen in (50/abs(ratio)) ways, as it needs to be a divisor of the ratio and lie in the range of (-50, -49, ..., -1).
So, the total number of favorable outcomes for negative common ratios is:
Σ(50/abs(ratio)) for ratio = -1 to -25.
Finally, the number of favorable outcomes for both positive and negative common ratios is the sum of the two.
Once we have the number of favorable outcomes, we can calculate the probability by dividing it by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes.
Now, you can calculate the probability using these calculations.
0.00122
What GP's are possible? If the 3 numbers are chosen independently, then we have
1,1,1
1,2,4
1,3,9
1,4,16
1,5,25
1,6,36
1,7,49
2,2,2
2,4,8
2,6,18
2,8,32
2,10,50
3,3,3
3,6,12
3,9,27
3,12,48
4,4,4
4,8,16
4,12,36
5,5,5
5,10,20
5,15,45
6,6,6
6,12,24
and so on.
If the three numbers are chosen without replacement, like picking three cards from a deck, then the number of possible GP's is much smaller.
Anyway, once you have the set of GP's you can figure the probability of each, then just add them up.