Probability

Consider a population of penguins, with the following absolute frequencies:

sex	female	male
Adelie	160	80
Chinstrap	20	80
Gentoo	80	80

Exercise 1: Probabilities

Calculate the probabilities that a randomly selected penguin:

• is female

• is an Adelie

• is a female Adelie

• is an Adelie or Chinstrap

• is male or female

Exercise 2: Conditional Probabilities

Calculate the probability that

• a randomly selected Adelie is a female.

• a randomly selected female is an Adelie

Exercise 3: Probability Tree

Draw a probability tree for the penguins. The first random event in the tree is the species, the second one is the gender. Calculate all probabilities and annotate the tree.

Exercise 4: Another Probability Tree

Now draw a second probability tree but this time the gender should be the first random event. Calculate all probabilities and annotate the tree with them.

Exercise 5: Total Probability

calculate the total probability p(Chinstrap) from the conditional probabilities p(Chinstrap|male) and p(Chinstrap|female).

Exercise 6: Bayes Theorem

Calculate the conditional probability p(Chinstrap|male). Apply the Bayes Theorem to also calculate p(male|Chinstrap).

Then apply the Bayes Theorem to calculate the conditional probabilities in the opposite direction. Do you obtain the same value for p(Chinstrap|male) that you started with?

Exercise 7: Independence

When picking random penguins from the above population, are the gender and the species independent random events?

Exercise 8: Something less amusing

In Germany, approximately 42000 of 900000 deaths in 2021 were due to lung carcinoma. 95% of lung carcinoma patients are former smokers. 25% of the entire population are smoking.

Estimate the probability that a smoker who died in 2022 died of lung carcinoma.

Assume that the probabilities did not change during one year (in a long-term projection one would have to take changes in smoking behavior, e.g. e-cigarettes into account).

(source: Wikipedia)

Exercise 9: Birthdays

Calculate the probability that two persons in the classroom have the same birthday. Check if it actually is the case.