Final answer:
The probabilities in a uniform distribution can be determined by the ratio of the interval length to the total distribution length. In this case, the probabilities for the given ranges are computed as simple fractions of the total span of the distribution and then converted to decimal form.
Step-by-step explanation:
The problem discusses a random variable X with a uniform distribution between −6°C and 6°C. The probabilities associated with different intervals of this uniform distribution can be computed using the properties of the uniform distribution.
- P(X < 0): Since the uniform distribution is symmetrical and spans from −6 to 6, half of the distribution lies below zero. Therefore, the probability that X is less than zero is 0.5 or 50%.
- P(−3 < X < 3): Here we're dealing with an interval that is half the width of the entire distribution and perfectly centered. This interval is ± 3°C from the center of the distribution, which means it encompasses exactly half of the distribution. The probability is therefore 1/2 or 0.50.
- P(−3≤ X ≤ 4): To find this, we calculate the length of the interval, which is 4 - (-3) = 7°C, and then divide by the total span of the distribution, which is 6 - (-6) = 12°C. The probability for this range is 7/12, which in decimal form is approximately 0.58 when rounded to two decimal places.
- P(k < X < k + 4): Since the distribution is uniform and the interval length is always 4°C, as long as the interval lies within the bounds of the distribution, the probability does not depend on the value of k. The interval length divided by the total length of the distribution gives a probability of 4/12 or 1/3, which in decimal form is approximately 0.33 when rounded to two decimal places.