At the beginning of winter, a homeowner estimates that the...

At the beginning of winter, a homeowner estimates that the probability is 0.4 that his total heating bill for the three winter months will be less than $$\$ 380$$. He also estimates that the probability is 0.6 that the total bill will be less than $$\$ 460$$. a. What is the probability that the total bill will be between $$\$ 380$$ and $$\$ 460$$ ? b. Given no further information, what can be said about the probability that the total bill will be less than $$\$ 400$$ ?

At the beginning of winter, a homeowner estimates that the probability is 0.4 that his total heating bill for the three winter months will be less than $$\$ 380$$. He also estimates that the probability is 0.6 that the total bill will be less than $$\$ 460$$.
a. What is the probability that the total bill will be between $$\$ 380$$ and $$\$ 460$$ ?
b. Given no further information, what can be said about the probability that the total bill will be less than $$\$ 400$$ ?

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Key Concepts

Cumulative Distribution Function (CDF)

The CDF is a fundamental concept in probability that gives the probability that a random variable is less than or equal to a specific value. It is used to determine probabilities over intervals by comparing values of the CDF at different points, which is especially useful when dealing with continuous random variables or when only certain cumulative probabilities are known.

Interval Probability Calculation

Interval probability is the method of finding the probability that a random variable lies within a certain range by taking the difference between the CDF values at the endpoints of the interval. For example, if the probability of the variable being less than the upper bound is known, and similarly for the lower bound, then the probability that the variable falls between these two is obtained by subtracting the lower CDF from the upper CDF.

Incomplete Information and Bounding Probabilities

When only limited information about a probability distribution is available, such as the probabilities up to certain thresholds, it is often possible to establish bounds for other probabilities even without a complete model. In such cases, one can only assert that the probability of an event lies within a certain range determined by the known cumulative probabilities, without being able to provide an exact value.

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