L5 · Sample size: why more trials give a better estimate
“I can explain why a relative frequency from many trials is a better estimate of the true likelihood than one from few trials, and I can choose which of two estimates to trust - and use the better one to predict.”
A fair coin's likelihood of heads is 1/2. A class recorded tosses in growing batches: after 4 tosses, heads came up 3 times - relative frequency 3/4 = 75%. After 100 tosses, heads came up 54 times - 54/100 = 54%. After 1,000 tosses, heads came up 498 times - 498/1000 = 49.8%. Step 1: Compare each with the true likelihood, 50%: the errors are 25 points, then 4 points, then 0.2 points. Step 2: With few trials, luck bounces the relative frequency far from the truth. With many trials, the lucky and unlucky streaks balance out, and the relative frequency settles close to the true likelihood.
The 4-toss result of 75% heads happened because...
A die is rolled to count 6s (true likelihood 1/6, about 16.7%). After 6 rolls, a 6 came up 3 times (relative frequency 1/2). After 600 rolls, a 6 came up 98 times (about 16.3%). Which relative frequency is the better estimate of 1/6? -> The 600-roll one (16.3% is very close to 16.7%; the 6-roll one is far away)
"As the number of trials grows, the relative frequency usually gets ___ to the true likelihood." -> closer
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.