Final answer:
The probability of selecting a prime number from 10 to 30 is 6/21, and the probability of selecting a perfect square or cube within the same range is 3/21.
Step-by-step explanation:
Calculating Probability of Primes and Perfect Powers
To find the probability that a randomly chosen number from 10 to 30 is a prime number, we list all the prime numbers within the range: 11, 13, 17, 19, 23, 29. There are 6 prime numbers out of 21 numbers total since there are 21 whole numbers from 10 to 30 (inclusive). Thus, the probability is 6/21 or about 0.2857.
For perfect squares and perfect cubes in the given range, the perfect squares are 16 and 25, and the only perfect cube is 27 (since 3³ = 27 and it lies within the range). However, since 27 is not a perfect square, we have a total of 3 numbers that fulfill the condition. The probability is therefore 3/21 or approximately 0.1429.