In a forest 20% of Mushrooms are Red, 50% Brown and 30% White. A Red mushroom is poisonous with a Probability of 20%. A mushroom is not Red is poisonous with Probability of 5%.

In a forest 20% of Mushrooms are Red, 50% Brown and 30% White. A Red mushroom is poisonous with a Probability of 20%. A mushroom that is not Red is poisonous with a Probability of 5%. What is the Probability that a poisonous mushroom in the forest is Red?

The probability that a poisonous mushroom in the forest is red is 0.5.

Given:

• 20% of mushrooms are red.
• 50% are brown.
• 30% are white.
• A red mushroom is poisonous with a probability of 20%.
• A mushroom that is not red is poisonous with a probability of 5%.

We want to find the probability that a poisonous mushroom in the forest is red.

Let:

• P(R) be the probability of picking a red mushroom (0.20).
• P(B) be the probability of picking a brown mushroom (0.50).
• P(W) be the probability of picking a white mushroom (0.30).
• P(Poisonous | R) be the probability that a red mushroom is poisonous (0.20).
• P(Poisonous | ¬R) be the probability that a mushroom that is not red is poisonous (0.05).

Using Bayes' theorem:

P(R | Poisonous) = (P(Poisonous | R) * P(R)) / P(Poisonous)

P(Poisonous) = P(Poisonous | R) * P(R) + P(Poisonous | ¬R) * (1 - P(R))

Now, let's calculate:

P(Poisonous) = (0.20 * 0.20) + (0.05 * (1 - 0.20))

P(Poisonous) = 0.04 + 0.04 = 0.08

Now, we can calculate:

P(R | Poisonous) = (0.20 * 0.20) / 0.08

P(R | Poisonous) = 0.04 / 0.08

P(R | Poisonous) = 0.5

So, the probability that a poisonous mushroom in the forest is red is 50%.

What is Probability Theory?

Probability theory is a branch of mathematics concerned with quantifying uncertainty and analyzing random phenomena. It provides a framework for understanding and reasoning about events or outcomes that are uncertain or subject to chance. The central concept in probability theory is the notion of probability, which assigns a numerical value between 0 and 1 to quantify the likelihood of a particular event occurring.

Key components of probability theory include:

1. Sample Space: The set of all possible outcomes of a random experiment. For example, when rolling a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

2. Events: Subsets of the sample space. An event is any collection of outcomes from the sample space. For instance, when rolling a die, the event "rolling an even number" corresponds to the subset {2, 4, 6}.

3. Probability: A numerical measure of the likelihood of an event occurring. It is typically denoted by P(E), where E is an event. Probabilities range from 0 (indicating impossibility) to 1 (indicating certainty). For example, the probability of rolling a 4 on a fair six-sided die is 1/6.

4. Probability Distribution: Describes the probabilities of each possible outcome of a random experiment. It may be discrete (for experiments with a countable number of outcomes, like rolling a die) or continuous (for experiments with an infinite number of outcomes, like measuring a person's height).

5. Random Variables: A variable that can take on different values with certain probabilities. It associates each outcome of a random experiment with a numerical value. Random variables can be discrete or continuous, depending on the nature of the outcomes they represent.

6. Probability Laws and Rules: Principles governing the behavior of probabilities, such as the law of large numbers, the multiplication rule, the addition rule, conditional probability, independence, and Bayes' theorem.

Probability theory finds applications in various fields, including statistics, physics, economics, finance, engineering, and computer science. It is fundamental for making decisions under uncertainty, modeling random phenomena, and analyzing data.