On the Likelihood of an Accident with Limited Trials

Introduction

Often, people are curious about the chances of a low-probability event occurring, such as an accident, when there are multiple attempts at it. This article aims to demystify the mathematics behind such scenarios, specifically focusing on the likelihood of an accident happening when the probability of the event in a single attempt is 1 in 10,000, and you make 1,000 attempts. Understanding this helps in making informed decisions and setting realistic expectations.

The Probability Calculation

Let's start by defining the event A as the occurrence of the accident in a given attempt. The probability of this event, ( P(A) ), is given as 0.0001 or 1/10,000. The question then becomes, what is the likelihood of this event happening at least once in 1,000 attempts?

To find the likelihood of not getting the accident in a single attempt, we use the formula:

[ P(text{no accident in one attempt}) 1 - P(A) 1 - 0.0001 0.9999 ]

The probability of not getting the accident in 1,000 attempts is then calculated as:

[ P(text{no accident in 1,000 attempts}) (0.9999)^{1000} ]

Using a calculator or computing tool, we find:

[ (0.9999)^{1000} approx 0.9048 ]

Hence, the likelihood of getting at least one accident in 1,000 attempts is:

[ P(text{at least one accident}) 1 - 0.9048 approx 0.0952 ]

This means that there is approximately a 9.52% chance of getting into at least one accident when you make 1,000 attempts.

The Role of Independence

It's important to note that each attempt is independent of the others, much like rolling a die multiple times. In such a scenario, the probability remains constant for each attempt. This is governed by the Law of Large Numbers, which asserts that over a large number of trials, the average of the results tends to get closer to the expected value.

So, if you roll a die, you have a 1 in 6 chance of getting a six on each roll. This probability does not increase with each roll; it remains constant. Similarly, with an accident occurring 1 in 10,000 times, the probability for each attempt is independent and remains the same.

The Law of Large Numbers vs. the Law of Small Numbers

While the Law of Large Numbers suggests that with a large number of trials, the actual outcomes will approach the expected probability, the Law of Small Numbers can sometimes lead to misunderstandings. For example, in the case of a 1 in 10 chance, you might think that after many attempts, you would have a 100% chance of getting the desired outcome, but this is not true. In fact, the probability remains 10% for each attempt, and the likelihood of never getting the outcome in 10 attempts is 0.9^10, which is approximately 0.348 or 34.8%.

Therefore, the chance of getting at least one successful outcome in 10 attempts is 1 - 0.348 or 0.652, which is around 65.2%.

Conclusion

Understanding the probability of events, especially in the context of multiple trials, is crucial for making informed decisions. The key takeaway is that for an independent event with a low probability, the likelihood of it occurring at least once over multiple attempts can be calculated using the complement rule. This article provides a clear example of how to apply these concepts, making the information accessible and easy to understand.