[hoist_failure] Clean up the code and correct minor errors

[HumphreyYang]

This PR cleans up the code and correct minor errors and typos across the lecture.

It also improves the printing and smooth some of the sentences.

[github-actions]

📖 Netlify Preview Ready!

Preview URL: https://pr-656--sunny-cactus-210e3e.netlify.app (a9df52c)

📚 Changed Lecture Pages: hoist_failure

[github-actions]

📖 Netlify Preview Ready!

Preview URL: https://pr-656--sunny-cactus-210e3e.netlify.app (174da1f)

📚 Changed Lecture Pages: hoist_failure

[jstac]

🤖 Status note for a future session — from a maintainer investigation on 2026-07-08 into why open-PR previews 404. Context only, not instructions.

Netlify preview: https://pr-656--sunny-cactus-210e3e.netlify.app/ currently returns 404.

Why previews are down (repo-wide findings)

1. This branch is stale — 166 commits behind main. A preview build compiles the whole site from this branch. This branch's lectures/house_auction.md still has unpinned !pip install prettytable, which now breaks on a wcwidth incompatibility. main fixed this on 2026-06-28 by pinning prettytable<3.18 (#939). This alone fails any rebuild of this branch until it's updated to main.

2. The arviz failure was a red herring — do NOT pin arviz or rewrite plotting. A 2026-07-07 rebuild also failed in ar1_bayes/ar1_turningpts with an arviz_plots figsize ValueError. That was a transient bug in an intermediate arviz-plots 1.x release, already fixed in arviz 1.2.0. Verified locally on a clean latest-stack venv: the real az.plot_trace(trace) cell (pymc + numpyro InferenceData) runs green. The lectures use only 1.x-compatible arviz APIs (plot_trace, summary, from_numpyro, compare).

Recommended first step for this PR

Update this branch to main (merge or rebase — pulls in #939 plus ~166 other commits), then let CI rebuild. On today's latest libraries the site builds clean, so the preview should return. house_auction is the known blocker; updating also picks up other since-merged fixes — rebuild and address any remaining per-lecture failures. Verify with:

curl -sI https://pr-656--sunny-cactus-210e3e.netlify.app/hoist_failure.html

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This PR touches: hoist_failure.md. Last CI build: success@2025-11-14. Branch: 166 commits behind main as of 2026-07-08.

[Chihiro2000GitHub]

Hi @HumphreyYang, I'm working through the NumPy random API migration in QuantEcon/meta#299, and hoist_failure.md is one of the lectures still on the list.

This PR already rewrites the sampling cell, so rather than open a separate PR that changes the same lines, I thought I would ask here first. While you are in there, would you be happy to use the Generator API for those draws?

rng = np.random.default_rng(1234)
s1 = rng.lognormal(μ, σ, n_samples)
s2 = rng.lognormal(μ, σ, n_samples)
s3 = rng.lognormal(μ, σ, n_samples)

That keeps the seed you added, and it saves us both from a conflict on the same lines.

If you would rather keep this PR as it is, I am happy to do the migration in a separate PR once this one is merged.