Question

The newspaper article "Folic Acid Might Reduce Risk of Down Syndrome" (USA Today, September 29 , 1999) makes the following statement: "Older women are at a greater risk of giving birth to a baby with Down Syndrome than are younger women. But younger women are more fertile, so most children with Down Syndrome are born to mothers under \(30 .\) " Let \(D=\) event that a randomly selected baby is born with Down Syndrome and \(Y=\) event that a randomly selected baby is born to a young mother (under age 30 ). For each of the following probability statements, indicate whether the statement is consistent with the quote from the article, and if not, explain why not. a. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) b. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.7\) c. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) d. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\) e. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.4\) f. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\)

Short answer

Only statement a is consistent with the quote from the article. All other statements (b, c, d, e, f) are inconsistent because they either have equal risk of Down Syndrome for both young and old mothers, or the likelihood of a baby being born to a young mother is less than that of an old mother.

Step-by-step solution

Analyzing Statement a

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.004, P(Y)=.7\). Here, the probability of Down Syndrome for babies born to younger mothers is less than the probability for babies born to older mothers, and the probability of a baby being born to a young mother is more than the baby being born to an older mother. Therefore, this is consistent with the quote.

Analyzing Statement b

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.001, P(Y)=.7\). Here, the probability of Down Syndrome for babies born to younger mothers is the same as the probability for babies born to older mothers. This conflicts with the quote that states the risk is higher for older women, so it is inconsistent.

Analyzing Statement c

The given statement shows \(P(D \mid Y)=.004, P\left(D \mid Y^{C}\right)=.004, P(Y)=.7\). Here again, the probability of Down Syndrome for babies born to younger mothers is the same as for babies born to older mothers. Since it conflicts with the fact that the risk is higher for older women, it is inconsistent.

Analyzing Statement d

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.004, P(Y)=.4\). Here, the probability of Down Syndrome for babies born to younger mothers is less than the probability for babies born to older mothers. However, the probability a baby being born to a young mother is less than the mother being older which contradicts the assumption in the article. Therefore, this is inconsistent.

Analyzing Statement e

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.001, P(Y)=.4\). In this case, the probabilities of Down Syndrome for babies born to younger and older mothers are the same which contradicts the article. Moreover, the total probability of a baby being born to a young mother is less than being born to an older mother. Thus, it is inconsistent.

Analyzing Statement f

The given statement shows \(P(D \mid Y)=.004, P\left(D \mid Y^{C}\right)=.004, P(Y)=.4\). Here, the probabilities of Down Syndrome for babies born to younger and older mothers are equal, and the total probability of a baby being born to a young mother is less than being born to an older mother. This statement is inconsistent with the article's assumptions.