Math Problem Statement
- A bus arrives at a station every day at a random time between 1:00 P.M. and 1:30 P.M. A person arrives at this station at 1:00 and waits for the bus. If at 1:15 the bus has not yet arrived, what is the probability that the person will have to wait at least an additional 5 minutes?
Solution
We are given that the bus arrives at a random time between 1:00 P.M. and 1:30 P.M. (uniformly distributed), and a person arrives at 1:00 P.M. If the bus hasn’t arrived by 1:15 P.M., we are asked to find the probability that the person will have to wait at least an additional 5 minutes.
Step-by-step solution:
-
Time interval:
The bus can arrive at any time between 1:00 P.M. and 1:30 P.M., meaning the bus arrives uniformly on the interval minutes (where 0 corresponds to 1:00 P.M. and 30 corresponds to 1:30 P.M.). -
Conditional time range:
At 1:15 P.M., we know that the bus has not arrived yet. This means the bus will arrive sometime between 1:15 P.M. and 1:30 P.M., corresponding to the interval minutes. -
Required event:
We are asked to find the probability that the person will have to wait at least 5 more minutes from 1:15 P.M., i.e., the bus arrives no earlier than 1:20 P.M. This corresponds to the bus arriving in the interval minutes. -
Uniform distribution: Since the bus arrives uniformly in the interval , the probability that it arrives after 1:20 P.M. (i.e., in the interval ) is given by the ratio of the length of the interval to the length of the interval .
- The length of the interval is minutes.
- The length of the interval is minutes.
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Final calculation: The probability is:
[ \text{Probability} = \frac{\text{Length of the interval [20, 30]}}{\text{Length of the interval [15, 30]}} = \frac{10}{15} = \frac{2}{3}. ]
Thus, the probability that the person will have to wait at least an additional 5 minutes after 1:15 P.M. is .
Would you like further details on any step or have additional questions?
Here are five related questions to explore:
- What is the probability that the bus arrives between 1:10 P.M. and 1:20 P.M.?
- How would the probability change if the bus were to arrive between 1:00 P.M. and 1:45 P.M. instead?
- If the person arrives at 1:05 P.M., what is the probability that they will wait less than 10 minutes?
- What is the expected waiting time for the person if they arrive at 1:00 P.M.?
- If the person arrives randomly between 1:00 P.M. and 1:30 P.M., what is the probability that they will have to wait more than 10 minutes?
Tip: When dealing with uniform probability distributions, the probability of an event is often simply the ratio of the favorable interval length to the total possible interval length.
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Math Problem Analysis
Mathematical Concepts
Probability
Uniform Distribution
Conditional Probability
Formulas
P(Event) = Length of favorable interval / Length of total interval
Theorems
Law of Total Probability
Basic Properties of Uniform Distribution
Suitable Grade Level
Grades 9-11
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