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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?


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Verified Answer :

Aliwohaish12

Draw a simple branch diagram to work the probabilities out.

You find that the chance of a poisonous mushroom is 0.08 and the chance of a red poisonous is 0.04.
So the probability that a poisonous mushroom is red is 1/2 or 0.5.

Using conditional probability, it is found that there is a 0.5 = 50% probability that a poisonous mushroom in the forest is red.

Conditional Probability

\(P(B|A) = \frac{P(A \cap B)}{P(A)}\)

In which

  • P(B|A) is the probability of event B happening, given that A happened.
  • \(P(A \cap B)\) is the probability of both A and B happening.
  • P(A) is the probability of A happening.

In this problem:

  • Event A: Poisonous.
  • Event B: Red

The percentages of poisonous are composed by:

  • 20% of 20%(red)
  • 5% of 80%(not red)

Hence:

\(P(A) = 0.2(0.2) + 0.05(0.8) = 0.08\)

The probability that is red and poisonous is:

\(P(A \cap B) = 0.2(0.2) = 0.04\)

The conditional probability is:

\(P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.04}{0.08} = 0.5\)

0.5 = 50% probability that a poisonous mushroom in the forest is red.

A similar problem is given at https://brainly.com/question/14398287

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