Three different numbers are chosen at random from the set {1,2,3,4,5}. The numbers are arranged in increasing order. What is the probability that the resulting sequence is an arithmetic sequence?
Since the order has to be increasing , the only possibilities are
1 2 3
2 3 4 , common difference of 1 for these three choices
3 4 5
1 3 5 , common difference of 2
(there are not enough numbers to have a common difference of 3)
The total number of triplets we can form are
5x4x3 = 60
Prob that our triplet forms an arithmetic sequence = 4/60 = 1/15
To find the probability that the resulting sequence is an arithmetic sequence, we first need to determine the total number of possible outcomes.
Since we are choosing three numbers from the set {1, 2, 3, 4, 5}, the total number of possible outcomes can be calculated using combinations. The number of combinations of selecting three elements from a set of five is given by:
C(5,3) = 5! / (3!(5-3)!) = 10
Now, let's determine the number of outcomes where the resulting sequence is an arithmetic sequence. To form an arithmetic sequence, the difference between consecutive terms should be the same.
If the common difference is 1, we have the following possibilities:
1, 2, 3
2, 3, 4
3, 4, 5
If the common difference is 2, we have the following possibilities:
1, 3, 5
2, 4, 5
Therefore, there are a total of 5 outcomes where the resulting sequence is an arithmetic sequence.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total possible outcomes:
Probability = Number of favorable outcomes / Total possible outcomes
= 5 / 10
= 0.5
So, the probability that the resulting sequence is an arithmetic sequence is 0.5 or 50%.