Question

two real numbers are selected independently at random from the interval [−20,10]. what is the probability that the product of those numbers is greater than zero?

Answer 1

To find the probability that the product of two real numbers selected from the interval [-20, 10] is greater than zero, we need to consider four cases: both numbers positive, both numbers negative, one positive and one negative, and one number zero. The probability is 1, meaning that it is certain that the product will be greater than zero.

Step-by-step explanation:

To find the probability that the product of two real numbers selected from the interval [-20, 10] is greater than zero, we need to determine the portion of the interval where the product is positive.

To have a positive product, both numbers need to have the same sign (either both positive or both negative).

Since the interval includes negative and positive numbers, there are four cases to consider:

1. Both numbers are positive.
2. Both numbers are negative.
3. One number is positive and the other is negative.
4. One number is zero (but the other can be either positive or negative).

The probability of each case can be calculated by determining the length of the interval where the condition is satisfied. Since the interval is [-20, 10], the length of the interval is 30.

Case 1: Both numbers are positive. The interval for this case is (0, 10]. The length of this interval is 10.

Case 2: Both numbers are negative. The interval for this case is [-20, 0). The length of this interval is 20.

Case 3: One number is positive and the other is negative. The interval for this case is (-20, 0) ∪ (0, 10]. The length of this interval is 30 (same as the total length of the interval).

Case 4: One number is zero. The interval for this case is [-20, 0) ∪ (0, 10]. The length of this interval is 30 (same as the total length of the interval).

Therefore, the probability that the product of the two numbers is greater than zero is equal to the sum of the probabilities of the four cases divided by the total length of the interval, which is 30.

Probability = (10 + 20 + 30 + 30) / 30 = 90 / 30 = 3/1 = 1

Answer 2

The probability that the product of two independently selected real numbers from the interval [-20,10] is greater than zero is 5/9, or approximately 0.5556.

Step-by-step explanation:

The goal is to determine the probability that the product of two randomly selected real numbers from the interval [-20,10] is greater than zero. For the product of two numbers to be positive, both numbers must be either positive or negative. Since the interval is [-20,10], we split it into two sub-intervals: [-20,0) for negative numbers and (0,10] for positive numbers.

First, we find the length of each sub-interval. The length of [-20,0) is 20, and the length of (0,10] is 10. The total interval length is 30. The probability that both chosen numbers are from the same sub-interval (and thus the product is greater than 0) is the sum of the probabilities of both selecting two positive numbers and selecting two negative numbers.

The probability of selecting two positive numbers is (10/30) × (10/30), and the probability of selecting two negatives is (20/30) × (20/30). We add these two probabilities to get the final answer:

P(product > 0) = (10/30) × (10/30) + (20/30) × (20/30) = 1/9 + 4/9 = 5/9 or approximately 0.5556.