What is the difference between direct fitness and inclusive fitness, and how are they calculated?
Direct fitness is the number of offspring an individual produces. Inclusive fitness is the direct fitness plus indirect fitness. Indirect fitness is calculated by the number of related individuals produced multiplied by the degree of relatedness to those individuals.
What is the degree of relatedness you have to your father's first cousin's child?
From you to your father is r=.5. From your father to his father is r=.5. From your grandfather to his sister (or brother) is r=.5. From your great aunt to her son is r=.5. And from your father's first cousin to his child is r=.5. Therefore your degree of relatedness is .5^5=.03125.
Your grandmother was dating three men when she became pregnant with your mother. One of the three men who might be your grandfather has another grandchild. What is your degree of relatedness to his grandchild?
The degree of relatedness through your grandfather is r=.5^4, but there is only a 1/3 probability that this man is your grandfather, so r=(1/3)(.5^4)=.0208.
In a haplodiploid honeybee colony, the Queen mates with three drones. The paternity of all members of the colony is unknown. Are you more likely to help your brothers or your sisters?
The queen is the only reproductive individual in the colony and so you know she is the mother of you, your sisters, and your brothers. Because males are unfertilized, you are only related to your brother through your mother, with r=.5^2=.25. You are related to your sisters through your mother with r=.5^2. Drone A has a 1/3 probability of being your father. He also has a 1/3 probability of being your sister's father. So your total relatedness to your sister is r=[.25 + (.5)^2(1/3)^2] = .28. You are more likely to help your sisters.
Is a tiger more likely to act altruistically toward a house cat or an elephant?
The tiger is unlikely to act altruistically toward either. Although both tigers and house cats are part of the large cat family, they are separated by millions of years. The degree of relatedness among individuals of separate species is far too insignificant to affect behavior.