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When a U.S. President is tried for impeachment, he will be acquitted if more than 1/3 of the Senators (currently 34) vote to acquit him. What is the minimal number of voters (as a percent of eligible U.S. voters) needed to keep these 34 Senators in office?

If these 34 Senators are from very small states, you don't need 1/3 of the national population to back a President. You don't even need everyone from these states, just more than half of those who bother to show up to the polls.

Because Republicans and Democrats have a different set of small states with Senators from that party, a good answer will provide different percentages for Republicans and Democrats.

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    If I understand the system correctly, then US senators are elected by a plurality in their state. So only 34 votes are required (provided all the other candidates poll 0 votes) My point is that the question is dependent on turnout, which can be almost arbitarily low. – James K Dec 5 '19 at 22:41
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    The question might be better posed if you considered the number of voters represented rather than the number voting, since as JamesK points out the latter is trivial. Though this number is also a bit trivial, as it would be the sum eligible voters (population would also be reasonable) from the smallest 1/3 of states. – Bryan Krause Dec 5 '19 at 23:55
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    @BryanKrause - For the record, the smallest 16 states have a combined population of 21,282,171, according to a July 1, 2019 estimate. That's pretty close to the estimated population of Florida, which is the third most populated state. So, the answer to this question is "Anywhere from just 16 people up to half of 21.3 million people". – Bobson Dec 5 '19 at 23:59
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    @jamesqf No, he doesn't, he's assuming literally only one person votes per state in each election. Whoever that guy votes for wins. Hypothetically even that's an overestimate on the minimum: if nobody votes at all, then it probably falls to the governor and/or legislature to decide who the senators are, so literally 0 votes could be cast directly for senator. – zibadawa timmy Dec 6 '19 at 5:27
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    @hszmv oh I see what you're talking about. That's a terrible criterion, directly in conflict with the policy on self-answered questions. There's no reason to exclude a question that is a good-faith effort to teach. – phoog Dec 6 '19 at 15:10
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Per my googling: 15,259,240 individuals is the absolute minimum number of voters required to prevent a conviction of the President in the Senate. That's sorting a list of the US states by population, then taking a simple majority of the population to guarantee the election. I did not check to see what the laws are regarding a plurality vs. a majority or a quorum by state. That means that this number may be higher under certain circumstances. Suppose that there was a spoiler campaign in a state where a candidate only needed (per the constitution) a plurality in order to be elected Senator. That would mean that fewer voters (probably just over 1/3) would be necessary to elect that individual.

I also assumed (in what is most likely a false assumption) that the whole population of the State would vote. For the 2018 midterm election, voter turnout was 53.4% -- the highest in a long, long time, but still only about half of all of the eligible population. If we assume a similar turnout to elect these senators (and that's a big assumption-- keep in mind that this most recent turnout was extremely abnormal), then the minimum required number of voters drops to 8,148,435 (I'm rounding up my last 0.16 person).

So there you have it. To absolutely positively be certain that you'll get the senators you want in office across 17 states, you need a minimum of 15,259,240 voters, appropriately partitioned by states. To probably get the senators that you want, you'd need to have 8,148,435 voters.

A quick note: The low population states are not all traditionally aligned with the same party. While some states, such as Alaska and West Virginia, are traditionally deep red, states like Rhode Island and Hawaii have been very blue in their voting. A scenario in which all 17 states nominated senators who agreed on something divisive such as the (implied) resistance to a legitimate impeachment is unlikely.

An edit upon thinking this over further: I reviewed total population, which includes individuals who are not eligible to vote. This answer could and should be revised to look at the number of eligible voters in the State, not simply the population.

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    Also worth noting that only a third of the senators get elected each time, which really messes with the math. – Bobson Dec 6 '19 at 18:46
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    True. And if memory serves, the re-election cycles for the two senate seats don't typically line up in most states, so most likely these votes would need to be controlled over a 4 year period. – NegativeFriction Dec 6 '19 at 18:47
  • @NegativeFriction They don't, by law. Only in cases of special elections would a state elect two senators at once, as I believe MN and GA did recently. – Azor Ahai Dec 11 '19 at 20:46
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The theoretical minimum is 9,410,010 voters (4.15% of U.S. voters) to (re-)elect the 34 Republican Senators needed to acquit a Republican President, and likewise 18,076,440 (7.96%) for Democratic Senators and President.

Analysis of Republican Senators Defending a Republican President

According to data from the U.S. Census Bureau, here are the states with Republican Senators, with the smallest number of voters (measured in thousands) in the November 2018 midterm elections. This table also lists the current number of Republican Senators from that state. Maine also has one Independent, who for the sake of analysis we will assume does not partake in an effort to protect a President; including him would not change the final answer.

I have also included the popular vote percent of each state for the last two Republican Presidential candidates. Many of these states are "deep red" states, meaning their voters strongly support Republican Presidents and Senators. Those states which are not "deep red" would be the logical targets for increased campaign support to protect their Republican Senators during re-election. Some of these Senators might lose, which would then require the next-smallest state, increasing the final answer -- but remember, this answer is the theoretical minimum.

                2018   Republican  2016 Trump  2012 Romney
State          Voters   Senators   Pop. Vote    Pop. Vote
-------------  ------  ----------  ----------  -----------
Wyoming          220       2         67.40%       68.64%
Alaska           263       2         51.28%       54.80%
South Dakota     331       2         61.53%       57.89%
North Dakota     335       2         62.96%       50.39%
Montana          518       1         56.17%       55.35%
Idaho            587       2         59.26%       64.53%
West Virginia    610       1         68.50%       62.30%
Nebraska         676       2         58.75%       59.80%
Maine            693       1*        44.87%       40.98%
Arkansas         919       2         60.57%       60.57%
Kansas          1152       2         56.65%       59.71%
Mississippi     1180       2         57.94%       55.29%
Utah            1214       2         45.54%       72.79%
Iowa            1335       2         51.15%       46.18%
Oklahoma        1350       2         65.32%       66.77%
Louisiana       1656       2         58.09%       57.78%
Kentucky        1746       2         62.52%       60.49%
Alabama         1830       1         62.08%       60.55%
South Carolina  1836       2         54.94%       54.56%
--------------  ----      --
TOTAL          18451      34

* plus one Independent

These 19 states have 34 Republican Senators, enough to acquit a President in an impeachment trial. Now suppose at least 51% of the voters in these states would stand by their President and re-elect their Republican Senators. (Technically it's 50% + 1, but I used 51%, which makes the answer slightly larger.)

51% of 18,451,000 voters is 9,410,010 voters. This is 2.88% of the overall U.S. population (327,167,434 per Census Bureau 2018 estimate) and 4.15% of citizens of voting age (227,019,486).

Analysis of Democratic Senators Defending a Democratic President

Here is the same analysis for Democratic Senators who would acquit a Democratic President:

               2018   Democratic  2016 Clinton  2012 Obama
State         Voters   Senators    Pop. Vote    Pop. Vote
------------  ------  ----------  ------------  ----------
Vermont         273       1*         56.68%       66.57%
Delaware        369       2          53.09%       58.61%
Rhode Island    403       2          54.41%       62.70%
Hawaii          427       2          62.22%       70.55%
Montana         518       1          35.75%       41.70%
New Hampshire   576       2          46.98%       51.98%
West Virginia   610       1          26.43%       35.54%
New Mexico      715       2          48.26%       52.99%
Nevada         1006       2          47.50%       52.36%
Connecticut    1370       2          54.57%       58.06%
Alabama        1830       1          34.36%       38.36%
Oregon         1918       2          50.07%       54.24%
Maryland       2320       2          60.33%       61.97%
Colorado       2342       1          48.16%       51.49%
Minnesota      2523       2          46.44%       52.65%
Massachusetts  2731       2          60.01%       60.65%
Wisconsin      2776       1          46.45%       52.83%
Arizona        2800       1          45.13%       44.59%
Washington     3234       2          52.54%       56.16%
Virginia       3319       2          49.73%       51.16%
New Jersey     3384       2          54.99%       58.38%
-------------  ----      --
TOTAL         35444      35  

* plus one Independent

The smallest states with Democratic Senators are not quite as small as the smallest Republican states, so their answer is going to be bigger than the Republican answer. Also, many of these states are not as "deep blue" as the Republican states are "deep red", so there is more of a chance that one of these Senators could be at-risk.

If I had included Independent Bernie Sanders in the analysis, it would have decreased the answer.

51% of 35,444,000 voters is 18,076,440 voters, which is 5.53% of the overall U.S. population and 7.96% of citizens of voting age.

Conclusions

Possible criticisms include:

  • Voter turnout is never the same. It may go up or down, increasing or decreasing the answer. However, the basic premise that these voters are a tiny percent of the overall U.S. would remain the same.
  • Only 1/3 of senators are up for re-election every two years. If a Presidential election occurs before a senator is up for re-election, the voters may no longer care about an impeachment vote anyway. This effect would tend to decrease the final answer.
  • I already mentioned that some of these states are swing states and may not support a Senator who acquits a President. You would need another Senator from a larger state, increasing the answer.
  • When calculating percent, everyone has a different opinion about which number should be in the denominator.

So you can argue about the precision of the answer, but the underlying conclusion that you only need a very small percent of voters to ensure re-election of the Senators who acquit a President is true. And it's true regardless of party; both Republicans and Democrats should be concerned by these results.

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    Your conclusion says something very different than your thesis, as written. The conclusion talks about re-electing senators who previously voted to acquit a president, but the rest of this seems to be based on what it would take to vote in those Senators beforehand. Separately, you can't just dismiss the six-year term. Both senators can't come from the same state if you're counting a single election's worth of voters, and if you're spreading it out over multiple elections, the question becomes meaningless. – Bobson Dec 6 '19 at 18:53
  • @Bobson: I fixed the question title to address your concern of elect vs. re-elect. As far as the 6-year term, the calculations presented here are for all 34 senators, not just those in a single election. – DrSheldon Dec 6 '19 at 19:05
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Theoretically, you could have one single person voting in each of 34 states, and these 34 people all voting for party A. And at the same time, in the other 16 states everyone without exception could vote for party B. The 34 voting for party A would be much less than 0.0001% of the voters.

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    While wildly unrealistic, this is not wrong. Which is why this isn't a useful question to begin with. – Bobson Dec 7 '19 at 22:51

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