A genetic trait is inherited in such a way that for a father with the trait, his male offspring have a 25% chance of inheriting it but his female offspring cannot inherit it. Suppose that in a certain population, 40% of offspring are male and offspring genders are independent events. For a father with the trait who has two offspring, what is the probability that the older offspring has the trait and the younger offspring does not?
I know this means that the older child must be male (40% chance) and there is then a 25% chance that he will have the trait. And the second child must be a daugther (60% chance), but am not sure where to go from here.
10% of offspring have the trait ... .25 * .40
four possible outcomes
... older yes, younger yes ... .10 * .10
... older yes, younger no ... .10 * .90
... older no, younger yes ... .90 * .10
... older no, younger no ... .90 * .90
offspring gender is a factor, but not directly in the calculation
56460
To solve this problem, we can break it down into two separate events: the older offspring having the trait and the younger offspring not having the trait. Let's calculate the probabilities for each event separately:
1. Probability that the older offspring is male: Since 40% of offspring are male, the probability of the older offspring being male is 0.40.
2. Probability that the older offspring has the trait: For a father with the trait, the male offspring have a 25% chance of inheriting it. Therefore, the probability that the older offspring has the trait (given that he is male) is 0.25.
3. Probability that the younger offspring is female: We already know that the older offspring is male, so now we need to calculate the probability that the younger offspring is female. Since 40% of offspring are male, the probability of the younger offspring being female is 0.60.
4. Probability that the younger offspring does not have the trait: According to the given information, the trait cannot be inherited by female offspring. Therefore, the probability that the younger offspring does not have the trait (given that she is female) is 1.
To find the probability that the older offspring has the trait and the younger doesn't, we multiply the probabilities of the two independent events:
Probability = (Probability of the older offspring being male) * (Probability of the older offspring having the trait) * (Probability of the younger offspring being female) * (Probability of the younger offspring not having the trait)
= 0.40 * 0.25 * 0.60 * 1
= 0.06
Therefore, the probability that the older offspring has the trait and the younger offspring does not is 0.06, or 6%.