Introduction to Probability and Event Occurrences

Probability is a measure of the likelihood of an event occurring. It is crucial in many fields, including statistics, scientific research, and even everyday decision-making. In the context of our discussion, we explore how many times an event with a 30% chance of occurring needs to take place to ensure a 90% probability of that event happening at least once. This piece delves into the theoretical and practical calculations required to answer this common question.

The Complement Rule and Its Application

The complement rule of probability is a fundamental concept used to find the probability of an event occurring by first determining the probability of the event not occurring. This rule states that the probability of an event not occurring, (P(A^c)), is (1 - P(A)). For an event with a 30% chance of occurring, we calculate the probability of this event not occurring and then use it to find how many times the event must occur to achieve a 90% probability of occurrence.

Step-by-Step Calculation

Determine the probability of non-occurrence:

If the event has a 30% chance of occurring, then the probability of it not occurring is (1 - 0.30 0.70) or 70%.

Calculate the probability of non-occurrence over multiple trials:

The probability that the event does not occur in n trials is given by 0.70^n.

Set up the equation for at least one occurrence:

To find the probability that the event occurs at least once, we use the complement rule: [1 - 0.70^n geq 0.90]

Reformulate the inequality:

This can be rearranged to: [0.70^n leq 0.10]

Solve for n:

To solve for n, we take the logarithm of both sides:

[n cdot log(0.70) leq log(0.10)]

Since the logarithm of 0.70 is negative, we divide both sides by it, which reverses the inequality:

[n geq frac{log(0.10)}{log(0.70)}]

Calculate the value of n:

[n approx frac{-1.000}{-0.155} approx 6.45]

Since n must be a whole number, we round up to the nearest whole number: [n 7]

Geometric Distribution and Summation of Probabilities

For a more detailed and rigorous approach, we can use the geometric distribution formula to sum the probabilities of the event occurring at different points in time until the cumulative probability meets or exceeds 90%. The geometric distribution gives the probability that the first occurrence of an event will happen on the n-th trial, given a constant success probability p. In this case, the probability of success p is 0.30. We need to sum the probabilities for n trials until the cumulative probability of at least one success is 90%, which can be represented as:

[sum_{n1}^N (1-p)^{n-1}p geq 0.90]

This sum converges quickly, and often only a few trials are needed to meet the threshold. The previous example showed that in 91.7% of cases, 7 or fewer trials are sufficient, which illustrates the efficiency of this method.

Approximate Calculations Using Logarithms

Using logarithms provides an intuitive and often simpler way to estimate the number of trials required. The formula for the smallest n can be approximated as:

[left(1 - e^{-frac{1}{p}}right)^{np} approx 1 - p^n approx 0.10]

For a 30% chance event, we can simplify the expression to:

[left(1 - e^{-frac{1}{0.30}}right)^{0.30n} approx e^{0.30n} approx 10]

This approximation is more straightforward to compute and provides a good ballpark estimate. Using the natural logarithm of 10 (approx. 2.302), we can estimate:

[0.30n approx 2.302]

[n approx frac{2.302}{0.30} approx 7.67]

While n 7 is the correct answer, this approximation is particularly useful when the probability p is very small. For example, if the probability is 3%, the approximation calculates:

[0.03n log(10) approx 0.03n times 2.302 approx 0.069n approx 2.302]

[n approx frac{2.302}{0.069} approx 33.34]

In conclusion, when working with rare events, the approximation provides a quick and accurate estimate. For more precise calculations, especially with more frequent events, the direct calculation using the complement rule or geometric distribution is most appropriate.

Conclusion and Application

Understanding how many times an event with a 30% chance of occurring needs to take place to have at least a 90% probability of happening can be crucial in various applications. From quality control in manufacturing to assessing risks in financial investments, knowing the number of trials needed to meet a certain threshold is essential. This piece has outlined the theoretical and practical methods to find this number, providing both precise calculations and useful approximations for practical applications. By grasping these concepts, you can make more informed decisions based on probabilistic outcomes.