Of all the fish in a certain river, 20 percent are salmon. Once a year, people can purchase a fishing license that allows them to catch up to 8 fish. Assume each catch is independent.

Of all the fish in a certain river, 20 percent are salmon. Once a year, people can purchase a fishing license that allows them to catch up to 8 fish. Assume each catch is independent. Which of the following represents the probability of needing to catch 8 fish to get the first salmon?

0.28

1/0.2

0.2(0.8)^7

0.8(0.2)

0.8(0.2)^7

The probability of needing to catch 8 fish to get the first salmon is 0.2(0.8)^7.

To find the probability of needing to catch 8 fish to get the first salmon, let's break down the process:

1. The probability of not catching a salmon on a single catch is 0.8 (since 20% of the fish are salmon, 100% - 20% = 80% of the fish are not salmon).
2. We need to catch 7 non-salmon fish in a row (since we're looking for the probability of needing to catch 8 fish to get the first salmon).

So the probability can be calculated as:

P=0.8^7

Therefore, the correct representation from the options is: 0.2(0.8)^7.

What is Probability Theory?

Probability theory is a branch of mathematics concerned with quantifying uncertainty and randomness. It provides a framework for understanding and analyzing random phenomena by assigning numerical measures, called probabilities, to various outcomes of an event. The foundational concepts of probability theory include:

1. Sample Space: This refers to the set of all possible outcomes of an experiment or random process. For example, when rolling a fair six-sided die, the sample space consists of the numbers {1, 2, 3, 4, 5, 6}.

2. Event: An event is any subset of the sample space. It represents a specific outcome or a combination of outcomes of an experiment. For instance, in the die example, the event of rolling an even number can be represented by the set {2, 4, 6}.

3. Probability: Probability is a numerical measure assigned to each event, indicating the likelihood of that event occurring. It is typically represented as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. For example, the probability of rolling a 4 on a fair six-sided die is 1/6.

4. Probability Distribution: This describes how probabilities are distributed over the various possible outcomes of a random experiment. It can be represented using mathematical functions, tables, or graphs. Common probability distributions include the uniform distribution, binomial distribution, normal distribution, and Poisson distribution.

5. Probability Rules: Probability theory includes several rules and principles that govern the manipulation and calculation of probabilities. These include the addition rule, multiplication rule, complement rule, and Bayes' theorem, among others.

Probability theory has widespread applications in various fields, including statistics, economics, engineering, physics, biology, and finance. It provides a rigorous framework for making decisions under uncertainty, modeling random phenomena, and analyzing data in uncertain environments.