Finding the probability of waiting at least five minutes

Key takeaway: This question is about a waiting-time probability, which is often modeled with an exponential distribution in statistics.

Identify the waiting-time model

The phrase “wait at least 5 minutes until the next hit” points to a waiting-time event. In many statistics problems, waiting times between random events are modeled by an exponential distribution. That means the probability of waiting at least a certain time is found with the survival function, not by subtracting a long table of intervals.

Here, the required probability is the chance that the wait lasts 5 minutes or more. The given value is 0.0067, which is the probability of that event.

Why the answer is the survival probability

For an exponential model, the probability of waiting at least time $t$ is

$P(T\ge t)=e^{-\lambda t}$

This quantity directly measures how likely it is that the next event has not yet happened by time $t$.

Because the question asks for at least 5 minutes, the relevant calculation is exactly this tail probability. If the problem statement or accompanying work already provides the numerical result, then the final probability is simply the stated value.

Final answer

The probability that we will have to wait at least 5 minutes until the next hit is

$\boxed{0.0067}$

Common interpretation issue

Students sometimes confuse “at least 5 minutes” with “within 5 minutes.” Those are complements. If a problem gives the probability of waiting less than 5 minutes, then the answer to “at least 5 minutes” would be $1 - P(T<5)$. Here, the value given is already the correct tail probability.

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Pitfalls the pros know 👇 A common mistake is to read “at least 5 minutes” as “exactly 5 minutes” or “within 5 minutes.” In continuous probability models, the probability of an exact time is typically 0, so the meaningful event is the interval tail. Another frequent error is using the complement in the wrong direction. If the event is phrased as waiting no less than 5 minutes, you need the right-tail probability, not the cumulative probability up to 5 minutes.

What if the problem changes? If the question were changed to “What is the probability that we will have to wait less than 5 minutes until the next hit?”, then the answer would be the complement of the tail probability. The new event is $P(T<5)$, so you would compute $1-0.0067=0.9933$ if 0.0067 is the probability of waiting at least 5 minutes. If the problem instead asked for “at most 5 minutes,” the interpretation would be the same in a continuous model: $P(T\le 5)=P(T<5)$.

Tags: exponential distribution, waiting-time probability, survival function