91Ó°ÊÓ

Fishing: Lake Trout At Fontaine Lake Camp on Lake Athabasca in northern Canada, history shows that about \(30 \%\) of the guests catch lake trout over 20 pounds on a 4 -day fishing trip (Source: Athabasca Fishing Lodges, Saskatoon, Canada). Let \(n\) be a random variable that represents the first trip to Fontaine Lake Camp on which a guest catches a lake trout over 20 pounds. (a) Write out a formula for the probability distribution of the random variable \(n\). (b) Find the probability that a guest catches a lake trout weighing at least 20 pounds for the first time on trip number \(3 .\) (c) Find the probability that it takes more than three trips for a guest to catch a lake trout weighing at least 20 pounds. (d) What is the expected number of fishing trips that must be taken to catch the first lake trout over 20 pounds? Hint: Use \(\mu\) for the geometric distribution and round.

Step by step solution

01

Understanding the Problem

This problem involves identifying the mathematical context: It is a geometric distribution, where each trip is an independent trial and the probability of catching a trout over 20 pounds is 0.3.

02

Formula for Probability Distribution

For a geometric distribution, the probability that the first success (catching a trout over 20 pounds) occurs on trip number \( n \) is given by the formula \( P(n) = (1 - p)^{n-1} \cdot p \), where \( p \) is the probability of success on each trial. Here, \( p = 0.3 \), so \( P(n) = (0.7)^{n-1} \cdot 0.3 \).

03

Probability of Catching on Trip 3

Substitute \( n = 3 \) into the formula from Step 1: \( P(3) = (0.7)^{3-1} \cdot 0.3 = (0.7)^2 \cdot 0.3 \). Calculate this to find that \( P(3) = 0.147 \).

04

Probability of More Than Three Trips

The probability that it takes more than three trips (i.e., on trip 4 or later) is given by \( 1 - P(1) - P(2) - P(3) \). We have \( P(1) = 0.3 \), \( P(2) = (0.7)^1 \cdot 0.3 = 0.21 \), and \( P(3) = 0.147 \). So the probability is \( 1 - 0.3 - 0.21 - 0.147 = 0.343 \).

05

Expected Number of Trips

For a geometric distribution with parameter \( p \), the expected number \( \mu \) of trials to get the first success is \( \mu = \frac{1}{p} \). Here, \( p = 0.3 \), so \( \mu = \frac{1}{0.3} = 3.33 \). Thus, on average, a guest will need to take about 3.33 trips.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution

In statistics, a probability distribution describes how the probability is distributed over the value of a random variable. In the case of a geometric distribution, we are interested in the probability of the first success occurring on a specific trial. For Fontaine Lake, each fishing trip is a trial. The success is catching a fish over 20 pounds. For a geometric distribution, the probability of having the first success on the nth trial is given by the formula: \[ P(n) = (1 - p)^{n-1} \cdot p \] where:

• \( p \) is the probability of success on each trial,
• \( (1 - p)^{n-1} \) is the probability of getting failures in all trials before the first success.

This formula reflects the nature of the scenario where each trial is independent, like each fishing trip to the camp. The guest catches the fish independently of previous trips.

Random Variable

A random variable is a variable representing a numerical outcome of a random process. In this context, the random variable \( n \) represents the first fishing trip at which a guest catches a trout over 20 pounds. The concept of a random variable helps in categorizing and analyzing the outcomes of probabilistic events using statistical methods. When discussing the geometric distribution, the random variable is often used to determine the number of trials needed for the first success. Understanding the random variable's role is crucial because it acts as a bridge between the real-world scenario (fishing trips) and mathematical probability theory.

Expected Value

The expected value measures the center of a probability distribution, giving us the average outcome we'd expect in the long run. For a geometric distribution, the expected number of trials to achieve the first success, \( \mu \), is calculated using the formula: \[ \mu = \frac{1}{p} \] where:

• \( p \) is the probability of a single successful trip (catching a trout over 20 pounds). With \( p = 0.3 \), the expected number of trips is \( \mu = \frac{1}{0.3} = 3.33 \).

This tells us that, statistically, a guest needs about 3.33 trips to catch a trout over 20 pounds. The expected value provides insight into the "average" scenario and is a powerful tool in predicting long-run behavior in probabilistic situations.

Trial Success

Trial success is a critical component of understanding geometric distributions. It refers to achieving the desired outcome in a single trial. In our example, a trial is successful when a guest catches a lake trout over 20 pounds. The probability of this happening on any given trip is \( p = 0.3 \). In a sequence of independent trials rarely affected by the outcomes of prior ones, the chance of success does not change. For the geometric distribution:

• A success on the first trial has a probability of \( 0.3 \).
• The probability of success remains constant at each subsequent trial.
• Failing at first only affects the likelihood of success occurring at a later stage.

This constancy ensures that each guest has the same chance with every new attempt, making the geometric model apt for scenarios like fishing at Fontaine Lake.