Gut check

A town is served by two hospitals. In the larger one, about 45 babies are born each day; in the smaller one, about 15 babies. Overall, about 50% of babies are boys, but the exact percentage varies from day to day.

For one year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital recorded more such days?

Estimate first

You flip a fair coin 100 times. What is the probability that you get exactly 50 heads?

Don't calculate — give your gut estimate.

%
Great intuition — it's about 8%. "Exactly 50" is the single most likely outcome, but it competes with 49, 51, 48, 52, … each almost as likely.
Lower than that — about 8%. Exactly 50 is the most likely single outcome, but it's just one of many plausible ones: 49, 51, 48, 52, … each nearly as likely.
Much lower — about 8%. "Fair coin" makes 50 feel inevitable, but exact equality is rare even when the coin is fair. You'll land between 40 and 60 about 95% of the time — just rarely on exactly 50.
Slightly higher — about 8%. You're right that it's small; 50 is still the single most likely count, it just shares probability with all its neighbors.

Explain your thinking

In a sentence or two: why does a larger sample make a sample percentage less variable?

Write a little first — committing to an answer before you look is what makes this useful.

One way to say it: Each individual observation is random, but in a large sample the random highs and lows mostly cancel each other out. For the overall percentage to land far from the truth, many independent observations would all have to lean the same way at once — and that gets less and less likely as the sample grows. (Formally: the standard error of a proportion shrinks in proportion to 1/√n.)