Final answer:
The probability that a randomly chosen number from 10 to 30 is a prime number is approximately 0.2857, and the probability that it is a perfect square or a perfect cube is about 0.1429.
Step-by-step explanation:
To find the probability that a number chosen at random from 10 to 30 is a prime number, we need to identify all the prime numbers within that range. The prime numbers are 11, 13, 17, 19, 23, and 29. Since there are 21 whole numbers between 10 and 30, the probability is 6/21 or approximately 0.2857.
The probability that a number is a perfect square or a perfect cube in the range of 10 to 30 is found by identifying such numbers. The perfect squares are 16 and 25, and the only perfect cube in this range is 27. Thus, there are 3 numbers that fit our criteria, so the probability is 3/21 or about 0.1429.