# Are first-borns more likely to attend Harvard?

Are first-born more likely to attend Harvard? Between 75% and 80% of students at Harvard are first-born. Do first-born children work harder academically, and so end up overrepresented at top universities? So claims noted philosopher Michael Sanded. But Antonym Milliner and Raphael Call find a simple fault in the statistical reasoning and give a more plausible explanation. Michael Sandal’s book Justices is a rewarding and accessible account of political philosophy. Based on a course he has taught at Harvard for over two decades, it contains an account of an interesting survey he has conducted tit his students.

Sanded was trying to demonstrate John Rally’s famous critique of three conceptions of distributive Justice – the feudal, the libertarian, and the meritocracy. These distribute worldly rewards according to three different criteria: birth, property, and genetically inherited natural ability, respectively. All three of these, claims Rails, are chance endowments, and are thus morally arbitrary. Only his difference principle, the idea that inequality is permissible only if it benefits the worst off in society, avoids this arbitrariness.

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One possible critique of Rally’s position is that t neglects the causal relationship between effort and achievement: those who work hard may achieve (and deserve) more of the world’s rewards. However, Rails argues that even effort may be a product of favorable, and arbitrary, individual circumstances. In order to support this point, Sanded asks his Harvard students to raise their hands if they are the first-born children in their families. According to Sanded, “About 75-80 percent raise their hands.

The result has been the same every time I have taken the poll. ” Sanded suggests that the fact that so many Harvard undergraduates are first born is evidence that birth order has a strong effect on the level of academic effort that a child exerts (assuming it has no effect on innate 2012 Antonym Milliner and Raphael Call academic abilities), and thus her chance of getting a place at Harvard. However, this reasoning is faulty. It is a classic case of base-rate neglect, a phenomenon widely studied in the psychology literature.

Whether 75-80% is a high or a low number depends on the proportion of first-born children amongst the offspring of mothers make reference to. Put another way, Sanded estimates he probability that you are a first-born, given that you are at Harvard, Pr(last-born I Harvard), when he is really interested in how the probability that you are at Harvard, given that you are nth-born, Pr(Harvard I enthrone), depends on n. The former quantity depends on the relative frequencies of nth-born children amongst mothers with children at Harvard, the “base rate”, and is thus not identifiable with the latter.

To get a handle on what Sandal’s data really imply about the relationship between birth order and the chance of attending Harvard, begin by noting that 41% of the children born in the US in 1991 (likely the birth ear of many of Sandal’s current students) were first born. There is thus a big gap between the national abstract and the proportion of first-born in Sandal’s class. If we assume that this gap is exclusively due to a birth order effect – first-born working harder – how large does this effect need to be to explain the data?

In order to answer this question, consider the following simple model. Given a population of mothers, consider the set of all their children. Let F be the subset of first-born jejunely First-born predominate at Harvard. But how large are Harvard families? 37 hillier in this set, and let H be the subset of Harvard students in this set. Let p(F I H) be the probability of being a first-born child, given that you are a Harvard student, and p(H I F) be the probability of being at Harvard, given that you are a first-born child. The Venn diagram in Figure 1 may help to visualize this.