According to a new report by the surgeon general

In underlined portion we have 'as' so choices with more are unidiomatic here , therefore C and D are out.
'had likelihood' is wrong as women doesn't posses any likelihood . Moreover , the idiom here should be 'as likely as ' , which is correctly used in option E .
Option E is correct.

please highlight had under the question

the correct structure is "as likely...as"

Hi USADream,

Thank you for your question. Let's tackle this question one problem at a time, and see if we can narrow it down to a correct answer.

Let's start with the obvious difference in how each answer starts: had three times the likelihood/were three times more likely. When we talk about how likely something is to happen, we say it's more/less likely to happen, not that a person possesses a likelihood (it's not really a thing you can own). Therefore, let's get rid of answers A & B because they are using the wrong idiom.

Next, let's look at how each answer sets up the comparison, and whether or not they use proper idiomatic structure and parallel structure. For comparisons, the idiomatic structure needs to look like one of these two:

more likely to X than Y
as likely to X as Y

To show both idiomatic structure and parallelism, let's look at each answer with the rest of the sentence added in at the end:

(C) were three times more likely to begin to smoke as women who went to college.
(WRONG = doesn't follow the "more likely to X than Y" format)

(D) were three times more likely that they would begin to smoke as women who went to college.
(WRONG = doesn't follow the "more likely to X than Y" format)

(E) were three times as likely to begin smoking as women who went to college.
(CORRECT = uses the proper "as likely to X as Y" format)

Therefore, we can determine that E is the correct answer.

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