Question

Tom went to the woods to gather some mushrooms, but he never bothered to learn anything about them. In the forest, 90% of mushrooms are delicious king boletus mushrooms, while 10% are highly poisonous death cap mushrooms.

1 (10 points). Tom is eating mushrooms one by one. What is the probability that he'll enjoy the first three but will get poisoned by the fourth one?
2 (20 points). If Tom eats 20 mushrooms, what is the expected number of death caps that he eats? What is the standard deviation?
3 (40 points). If Tom eats 20 mushrooms, what is the probability that he will eat 0 death caps? Less than 2 ?
4 (30 points). Tom gathered 20 mushrooms and wants to choose 5 of them for cooking. How many ways are there to do it?
5 (30 points). Tom gathered lots of mushrooms, divided them into a few plastic bags, and put them in his freezer. Each bag has 1 death cap in it on average. What is the probability that the first bag he opens will contain no death caps?
6(20 points). Tom gathered 100 mushrooms. What is the z-score of 20 being death caps?

Answer 1

The student's question involves using probabilities, combinations, and distributions to determine outcomes related to mushroom picking, including risk of poisoning, expected values, and more. The calculations are based on known percentages of edible and poisonous mushrooms.

Step-by-step explanation:

The student has asked about the probability of various events occurring when randomly picking mushrooms, some of which are poisonous. The questions involve calculations based on the percentage of edible vs. poisonous mushrooms, combination calculations, probability distributions, expected values, standard deviations, and z-scores.

1. Probability of eating poisonous mushroom fourth

The probability that Tom will enjoy the first three but get poisoned by the fourth mushroom is the product of the probabilities of these independent events. (0.9 x 0.9 x 0.9 x 0.1) = 0.0729 or 7.29%.

2. Expected number and standard deviation of death caps

The expected number of death caps Tom eats after eating 20 mushrooms is 20 x 0.10 = 2 death caps. The standard deviation σ is found using the binomial distribution formula σ = √(np(1-p)), which gives σ = √(20 x 0.10 x 0.90) ≈ 1.342.

3. Probability of eating 0 or less than 2 death caps

The probability that Tom will eat 0 death caps after 20 mushrooms is (0.9^20) ≈ 0.1216 or approximately 12.16%. For less than 2 death caps, we sum the probabilities for 0 and 1, using binomial coefficient formulas.

4. Number of ways to choose 5 mushrooms from 20

Tom can choose 5 mushrooms from 20 in (20 choose 5) = 15504 ways. This is calculated using the combination formula C(n, r).

5. Probability first bag contains no death caps

Given that each bag contains on average 1 death cap, the scenario can be approximated with a Poisson distribution. The probability of a bag containing no death caps is e^(-1), which is approximately 0.3679 or 36.79%.

6. Z-score for 20 death caps out of 100

The z-score is calculated using the formula Z = (X - μ) / σ. For 100 mushrooms with 10% death caps, the mean μ is 10 and standard deviation σ is √(100 x 0.1 x 0.9) = 3. Therefore, the z-score for 20 death caps is (20 - 10) / 3 ≈ 3.33.