Consider two events \(E_{1}\) and \(E_{2}\), where \(P(E_{1}) = 0.3\) and \(P(E_{2}) = 0.5\). Calculate the following probabilities:
In a population that is in Hardy-Weinberg equilibrium, \(P(a) = 0.1\) and \(P(A) = 0.9\). Find the probability of each possible genotype.
For Activity 1, calculate \(P(E_{2} | E_{1})\) assuming \(P(E_{1} | E_{2}) = 0.35\).
In this case, are these two events independent?
Suppose the probability of being affected by H1N1 flu is 0.02. We found that among people who are affected by H1N1, the probability that a person washes her hands regularly is 0.3. If the probability of washing hands regularly in general (regardless of whether the person has the H1N1 flu or not) is 0.6. What is the probability of getting the H1N1 flu if a person washes her hands regularly?
A person has received the result of his medical test and realized that his diagnosis was positive (affected by the disease). However, the lab report stated that this kind of test has false positive rate of 0.06 (i.e., diagnosing a healthy person, \(H\), as affected, \(D\) ), and false negative rate of 0.038 (i.e., diagnosing an affected person as healthy). Therefore, while this news was devastating, there is a chance that he was misdiagnosed. After some research, he found out that the probability of this disease in the population is \(P(D) = 0.02\). Find the probability that he is actually affected by the disease given the positive lab result.
Assume that the the probability of having the disease is 0.4, and the disease is not genetic (i.e., it is independent from the genotype of individuals). Also assume that the gene \(A\) has two alleles \(A\) and \(a\) such that \(P(A) = 0.3\) and \(P(a) = 0.7\). If the population is in Hardy-Weinberg equilibrium, write down the sample space for the combination of the disease status (\(D\) for diseased and \(H\) for healthy) and different genotypes along with the probability of each possible combination.
For the above question, find the probabilities for all possible combinations of genotypes and the disease status assuming that the disease is related to the gene {} such that \(P(D| aa) = 0.5\) and \(P(D | Aa) = P(D | AA) = 0.3\).
Suppose a pregnant woman is going to give birth to a girl or a boy with equal probabilities. However, if the baby is a boy, the probability that he has black (Bk) hair is 0.7, whereas this probability is 0.4 if the baby is a girl. Alternatively, the baby could have blond (Bd) hair. Using a tree diagram, find the sample space and the corresponding probabilities for all possible combinations of gender and hair color for the baby.