When captured in the Eyrie, why didn't Tyrion just wait it out rather than risk his life in a trial by combat with a stranger as champion?

Question

When captured in the Eyrie, Tyrion himself called for the trial by combat; it wasn't foisted upon him by his captors. Why would he take this extremely risky move when he could instead have kept waiting for his powerful family to rescue him?

Tyrion may have assumed that his sword master brother would be able to stand as his champion, but this clearly was a miscalculation because his captor Lysa Arryn refused to delay the trial by the many days or weeks it would take for his brother to arrive. Even then, Tyrion continued with the trial by combat. When the sellsword Bronn agreed to stand as Tyrion's champion, Tyrion accepted this champion despite him being a stranger.

Answer 1

I don't remember what's in the tv-show, but the books make it pretty clear that his life is in danger in the sky cells, that people fall off them to their death, and the gaoler Mord has stolen his cloak, doesn't feed him and regularly beats him, and acts like he tries to push him off the edge of the cell. So Tyrion tricks Mord to get him an audience to "confess" in order to save his life from death by exposure and torture hidden away in a dungeon, and bring him in front of witnesses.

A quote from the chapter highlights this.

At first he had consoled himself that this imprisonment could not last long. Lysa Arryn wanted to humble him, that was all. She would send for him again, and soon. If not her, then Catelyn Stark would want to question him. This time he would guard his tongue more closely. They dare not kill him out of hand; he was still a Lannister of Casterly Rock, and if they shed his blood, it would mean war. Or so he had told himself.

Now he was not so certain.

Perhaps his captors only meant to let him rot here, but he feared he did not have the strength to rot for long. He was growing weaker every day, and it was only a matter of time until Mord's kicks and blows did him serious harm, provided the gaoler did not starve him to death first. A few more nights of cold and hunger, and the blue would start calling to him too.

[....]

He would have to get out of here, and soon. His chances of overpowering Mord were small to none, and no one was about to smuggle him a six-hundred-foot-long rope, so he would have to talk himself free. His mouth had gotten him into this cell; it could damn well get him out.

Tyrion pushed himself to his feet, doing his best to ignore the slope of the floor beneath him, with its ever-so-subtle tug toward the edge. He hammered on the door with a fist. "Mord!" he shouted. "Turnkey! Mord, I want you!"

Answer 2

It is all pretty much laid out in the scene when Trion demands the trial. After some time in the skycells, Tyrion says that he will confess his crimes. However, this is more a ruse to just even obtain an audience with the Lords and Ladies of the Vale. He goes on admitting to a bunch of sultry "crimes". But when challenged by Catelyn Stark that he stand accused of hiring an assassin he denies this.

At this point Lady Lysa Arryn commands that Tyrion be return to his cell and Tyrion realizes he will get no chance of justice and who knows when Lannister reinforcements may arrive (if they even know where he is).

[Tyrion] Is this how justice is done in the Vale? You accuse me of crimes, I deny them. So you throw me into a cell to freeze and starve. Where is the King's justice? I am accused and demand a trial.

[Lysa] If you are tried and found guilty then by the King's own laws you will pay with your life.

[Tyrion] I understand the law.

[Lysa] We have no executioner in the Eyrie. Life is more elegant here. Open the moon door.

This is when Tyrion realizes he will not have any real trial so opts for his next best option, a trial by combat...

Answer 3

The sky cells are highly uncomfortable and potentially deadly. Rather than freeze and starve, he hatched the plan to demand a trial by combat, as mentioned above. In the book:

"I thank you, my good lady, but I see no need to trouble Lord Robert," Tyrion said politely. "The gods know the truth of my innocence. I will have their verdict, not the judgment of men. I demand trial by combat."

But that was not the whole of his plan. He had a plan to win a trial by combat:

"And now I demand a champion, such as you have chosen for yourself. My brother Jaime will gladly take my part, I know."

"Your precious Kingslayer is hundreds of leagues from here," snapped Lysa Arryn.

"Send a bird for him. I will gladly await his arrival."

Jaime is one of the foremost swordsmen in the land, and known for it. He would have been an ideal champion, but Lysa wouldn't have it, which I'm actually not sure if she legally could deny his chosen champion:

"You will face Ser Vardis on the morrow."

"Singer," Tyrion said, turning to Marillion, "when you make a ballad of this, be certain you tell them how Lady Arryn denied the dwarf the right to a champion, and sent him forth lame and bruised and hobbling to face her finest knight."

"I deny you nothing!" Lysa Arryn said, her voice peeved and shrill with irritation. "Name your champion, Imp . . . if you think you can find a man to die for you."

"If it is all the same to you, I'd sooner find one to kill for me." Tyrion looked over the long hall. No one moved. For a long moment he wondered if it had all been a colossal blunder.

Then there was a stirring in the rear of the chamber. "I'll stand for the dwarf," Bronn called out.

He suspected Bronn might fight for him, and he knew how deadly he was on the way up, fighting through the mountain men.