Final answer:
To find the probability that Ella randomly plants the trees so that the two types of trees alternate, we need to calculate the ratio of the number of arrangements where the trees alternate to the total number of arrangements. Using permutation calculations, the probability can be found by taking the ratio of the number of ways to arrange the two groups of trees to the total number of possible arrangements.
Step-by-step explanation:
In order to determine the probability that Ella randomly plants the trees so that the two types of trees alternate, we need to consider the total number of ways the trees can be arranged and the number of ways they can be arranged such that they alternate.
Since there are 6 sycamores and 6 palm trees, there are a total of 12 trees. The total number of ways to arrange these trees is 12!.
Now, let's consider the number of ways the trees can be arranged such that they alternate. We can treat the two types of trees as distinct groups and consider the number of ways to arrange these two groups. There are 2! ways to arrange the groups. Within each group, there are 6! ways to arrange the trees.
Therefore, the probability that Ella randomly plants the trees so that the two types of trees alternate is equal to (2! * 6! * 6!) / 12!.
Calculating this value will give you the desired probability. Remember to round your answer to four decimal places.