12:11 a.m. (Wed) CBS News projects that Ted Cruz wins Idaho.

10:07 p.m. Kasich, at an election night rally in Columbus, Ohio, celebrated his stronger-than-expected finish in Michigan Tuesday night and reiterated his confidence that he'll win his home state of Ohio next Tuesday.

"We're very pleased with what's happened, and when you think about where we've come in the great state of Michigan--from basically single digits where people said, 'Well, how's he going to do,' to where we are in all likelihood going to finish in second place," he said.

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"We're all familiar with March Madness, and now the home-court advantage is coming north. And next week we're going to win the state of Ohio," Kasich continued, to cheers.

10:00 p.m. Ahead of results in the Idaho primary and Hawaii caucuses, here's a reminder of how many states each GOP candidate has won so far:

9:30 p.m. At a news conference in Jupiter, Fla., Trump celebrated his twin victories in Michigan and Mississippi, noting the efforts to take him down have not been successful.

"I don't think I've ever had so many horrible, horrible things said about me in one week--$38 million worth of horrible lies, but that's okay," he said. "It shows you how brilliant the public is, because they knew they were lies."

"To get these kind of numbers, where they call them immediately, is just something very special," he continued. "So I want to thank the public, I want to thank the people of Michigan, I want to thank the people of Mississippi."

9:18 p.m. According to the CBS News exit poll, Trump ran strongly among men (he won 44 percent) but did not do well among women: he took 28 percent, versus 29 percent for Cruz and 26 percent for Kasich.

Trump also ran well among Republican voters without a college degree (he took 44 percent, compared with 26 percent for Cruz and 19 percent for Kasich). Kasich, however, edged both Cruz and Trump among voters with a college degree (30 for Kasich, 27 percent for Trump, and 26 percent for Cruz).

Cruz ran strongest among those who said they are "very conservative" and among evangelical Christians. However, Trump did equally well among evangelicals and much better among those who said they were not evangelical Christians.

Thirty-eight percent of Republican voters said that they made up their minds in the last week--and Kasich benefited the most from those voters. He received support from 42 percent of them, compared with 25 percent for Cruz, 20 percent for Trump, and just 8 percent for Rubio. Trump was the clear winner among those who said they made up their minds earlier.

What would have happened if only Trump and Cruz were on the ballot? Forty-six percent said that they would have voted for Cruz, 37 percent for Trump, and 12 percent said they wouldn't have voted. Even though Rubio ran a distant fourth overall, in a two-way race with Trump, 44 percent said they would have voted for Rubio and 43 percent would have backed Trump.

9:06 p.m. CBS News projects that Donald Trump wins Michigan.

9:00 p.m. CBS News projects that Donald Trump is leading the Republican field in Michigan.

8:43 p.m. One more interesting tidbit out of the Mississippi GOP exit polls: the survey asked voters who they would have supported if only Trump and Cruz had been on the ballot.

Forty-nine percent said they would have voted for Trump, compared with 44 percent who said they would have voted for Cruz. Six percent said they wouldn't have voted.

In a two way race between Trump and Rubio, Trump would have crushed Rubio by 60 percent to 29 percent.

8:37 p.m. CBS News projects that Donald Trump has won Mississippi.

8:24 p.m. According the CBS News exit poll, Trump beat Cruz among virtually all demographic groups in Mississippi.

Among men: Trump 49 percent to Cruz 34 percent

Among women: Trump 41 percent to Cruz 37 percent

Among those under 45: Trump 41 percent to Cruz 38 percent

Among those over 45: Trump 47 percent to Cruz 35 percent

He even beat Cruz among those with a college degree: 42 percent to 34 percent; he only did a little better among those with less than a college degree (Trump 47 percent to Cruz's 38 percent).

White evangelical Christians made up 76 percent of Republican primary voters, and Trump topped Cruz 45 percent to 40 percent. Cruz ran most strongly among those primary voters who said they were "very conservative," beating Trump 51 percent to 40 percent. However, Cruz only won 26 percent of those who said they were "somewhat conservative" and 19 percent of those who said they were "moderate."

As has been the case in previous primaries this year, Trump did much better among those voters who said they made up their minds over a week ago: he took 54 percent of those voters, compared with Cruz's 31 percent. Trump ran much worse among those who said they made up their minds in the last week (26 percent to 44 percent for Cruz and 16 percent for Kasich). Only 8 percent of the late deciders voted for Rubio.

8:00 p.m. CBS News projects that Donald Trump is leading the Republican field in Mississippi. Cruz is competing with Trump for the top spot; Rubio and Kasich are far behind.

7:30 p.m. With the first polls closing in half an hour, here's a reminder of where the candidates stand in the delegate fight so far:

6:25 p.m. More than a third of Michigan Republican voters chose their candidate in the last week, according to exit polling:

6:15 p.m.Here's how many delegates are at stake tonight in the Republican race:

5:50 p.m.Some more insights from early exit polling:

Just over half of Republican primary voters in Michigan and Mississippi think that trade with other countries takes away U.S. jobs; only about a third believe that it creates U.S. jobs.

Republican primary voters in Michigan and Mississippi are divided over whether illegal immigrants working in the US should be offered a chance to apply for legal status or deported back to their home county. In Mississippi,voters are almost evenly split on this issue: 51 percent want them deported; 46 percent say they should be offered legal status. Voters in Michigan are more supportive of offering legal status (56 percent) vs 37 percent who want them deported.

Large majorities of Republican voters in Michigan (62 percent) and Mississippi (75 percent) support temporarily banning Muslims who are not US citizens from entering the country.

Significant numbers of Republican voters in Michigan (37 percent) and Mississippi (31 percent) say that they made up their minds who to vote for in the last week.

5:15 p.m.CBS News' first exit poll results are coming in from Mississippi and Michigan.

Among Republican voters in both states, the economy and government spending are the top two issues. When it comes to candidate qualities, voters in Michigan said the most important is a candidate who shares their values, while in Mississippi voters' top quality was a candidate who can "bring change."

In both states, GOP voters are looking for an outsider candidate: asked whether they prefer a candidate who is outside politics or has political experience, 60 percent of voters in Mississippi and 52 percent in Michigan said they prefer an outsider.

5 p.m.We're back for another primary night, with four states holding nominating contests Tuesday. Both Democrats and Republicans will head to the polls in Michigan and Mississippi, while just Republicans hold contests in Idaho and Hawaii.

After retired neurosurgeon Ben Carson's exit from the race last week, Republicans are down to four candidates: businessman Donald Trump, Texas Sen. Ted Cruz, Florida Sen. Marco Rubio and Ohio Gov. John Kasich.

The contest most political observers are keeping an eye on tonight is Michigan, which has been competitive on both sides and will test the staying power of front-runner Trump. Both parties have held debates in the state in the last week: Democrats met in Flint on Sunday night, while Republicans faced off in Detroit last Thursday.

According to CBS News' latest count, Trump leads with 378 delegates of the 1,237 required to secure the nomination, followed by Cruz with 298, Rubio with 146 and Kasich with only 35.

The first polls of the night will close in Mississippi at 8 p.m. ET. Michigan is next at 9 p.m. ET, followed by Idaho at 11 p.m. ET and Hawaii at 1 a.m. ET.

Stony Brook University political science professor Stanley Feldman contributed to this story.