Final answer:
The conditional probability P(x < 3.5|x < 4) in a uniform distribution X ~ U(1.5, 4.5) is calculated as the ratio of lengths on the interval, resulting in a probability of 0.8 or 80%.
Step-by-step explanation:
To find the conditional probability P(x < 3.5|x < 4) given that X follows a uniform distribution X ~ U(1.5, 4.5), we must consider only the outcomes where x is less than 4. Since the uniform distribution is continuous and flat, the conditional probability is the same as the ratio of the lengths on the interval.
In this case, the interval from 1.5 to 3.5 is 2 units long and the interval 1.5 to 4 is 2.5 units long. Therefore, the conditional probability is the ratio of these two lengths, which is:
P(x < 3.5|x < 4) = (3.5 - 1.5) / (4 - 1.5) = 2 / 2.5 = 0.8
This means there is an 80% chance that x is less than 3.5 given that it is already less than 4.