Problem 9 - An Introduction to Mathematical Statistics and Its Applications

Question

Astronomers estimate that as many as one hundred billion stars in the Milky Way galaxy are encircled by planets. If so, we may have a plethora of cosmic neighbors. Let \(p\) denote the probability that any such solar system contains intelligent life. How small can \(p\) be and still give a fifty-fifty chance that we are not alone?

Step-by-Step Solution

Answer

The smallest value of \(p\) that gives a 50-50 chance of not being alone in the Milky Way is \(p = 1-e^{\frac{ln(0.5)}{100,000,000,000}}\).

Step 1: Define the problem in terms of probability

We need to find the probability \(p\) such that there is a 50% chance that at least one of the 100 billion stars has intelligent life. The problem can be modeled as a Bernoulli distribution trial, where each trial represents a solar system and the positive result indicates presence of intelligent life. The Bernoulli distribution is defined as \(P_{X=k}=p^k(1-p)^{1-k}\) , where \(P_{X=k}\) is the probability of \(k\) successes (if \(k=1\)) or \(k\) failures (if \(k=0\)), and \(p\) is the probability of one success. Here, a success means finding intelligent life.

Step 2: Use the complementary rule

Since the objective is to find the probability of at least one success, it's easier here to work with its complement - the situation where there are no successes across all trials. The probability of zero successes across 100 billion stars, based on our Bernoulli model, is \((1-p)^{100,000,000,000}\) . But we want this probability to be the complement of 50%, i.e., the other 50%. So, we want to solve the equation \((1-p)^{100,000,000,000} = 0.5\) .

Step 3: Solve for \(p\)

Solving the above equation for \(p\), we see that it is an exponential equation. Taking the natural log of both side we get \(100,000,000,000*ln(1-p) = ln(0.5)\) or \(ln(1-p) = \frac{ln(0.5)}{100,000,000,000}\). Exponentiating both sides, we get \(1-p = e^{\frac{ln(0.5)}{100,000,000,000}}\). Solving for \(p\), we find \(p = 1-e^{\frac{ln(0.5)}{100,000,000,000}}\).

Key Concepts

Bernoulli Distribution

In probability theory, the Bernoulli distribution is a simple yet essential concept.It models situations where there are two possible outcomes, often termed success and failure.In this case, think of each star system as a trial of a Bernoulli distribution. Here, "success" means discovering intelligent life, while "failure" means no intelligent life is found.

• Each trial can result in either success (with probability \(p\)) or failure (with probability \(1-p\)). This dichotomy makes the Bernoulli distribution ideal for modeling binary events.
• The distribution is defined by just one parameter, \(p\), which is the probability of success for each individual trial.

In the scenario of our galaxy, we consider 100 billion trials (one for each solar system). The probability of finding intelligent life at least once is what we're trying to determine.Understanding this distribution helps us model binary outcomes in many scientific and practical contexts, from coin flips to finding life beyond Earth.
By framing the problem this way, we simplify understanding of how to estimate probabilities across vast scales.

Complementary Probability

Complementary probability is a strategic concept in probability theory.Instead of calculating the desired probability directly, it's often easier to compute the complementary event's probability and then subtract it from one.

• For example, to find the chance of at least one success, it's quicker to first determine the probability of no successes at all.
• The idea is captured by the simple formula: \(P(A) = 1 - P(A^c)\), where \(P(A^c)\) is the probability of the complementary event.

In our problem, this means calculating the chance that none of the 100 billion stars have intelligent life - this would be a complete failure.So, if this failure probability is 50%, then the complementary probability, or our success chance, is also 50%. This trick reduces complexity in solving many probability problems.
Complementary probabilities offer a streamlined path to solve probability challenges, especially when working with large data sets or many trials.

Exponential Equation

The exponential equation is a powerful tool in mathematics, often used to model growth and decay processes.Solving exponential equations is vital in many scientific fields, including physics, biology, and computing.

• An exponential equation contains a variable in the exponent, such as our equation \((1-p)^{100,000,000,000} = 0.5\).
• To solve, we often use the property of logarithms to bring the variable to a manageable form. This involves taking the logarithm of both sides.

For the given problem, taking the natural logarithm helps transform the complex exponent into an easily solvable linear form:\[100,000,000,000 \cdot \ln(1-p) = \ln(0.5)\]
This step reduces the problem to basic algebra, where we solve for \(p\).
The relationship between exponential and logarithmic forms is key; it allows us to express the probability that answers the cosmic question of finding intelligent life.
Understanding exponential equations lets us tackle complex evolutions of systems, allowing for precise modeling and clearer insights into possible future scenarios.