In 30 seconds
3Blue1Brown uses three sellers’ ratings to introduce binomial probabilities and the distinction between likelihood and inference. The video previews Laplace’s rule of succession and explains that the rule depends on modeling assumptions.
- 1Sample size mattersMore observations can increase confidence even when the observed percentage is lower.
- 2Laplace’s ruleAdding one positive and one negative gives a model-dependent prediction for the next experience.
- 3The binomial formulaIndependent trials with success rate s yield C(n,k)s^k(1−s)^(n−k).
- 4Likelihood is not a posteriorThe probability of data given a rate is different from the probability of a rate given data.
- 5Assumptions matterThe rule depends on a prior and on assumptions such as independence and a stable rate.
Takeaways
- More observations can increase confidence even when the observed percentage is lower.
- Adding one positive and one negative gives a model-dependent prediction for the next experience.
- Independent trials with success rate s yield C(n,k)s^k(1−s)^(n−k).
- The probability of data given a rate is different from the probability of a rate given data.
- The rule depends on a prior and on assumptions such as independence and a stable rate.
Read the edited text by chapter
Comparing seller ratings
The presenter introduces the data and previews a model-dependent rule.
Three sellers offer the same product. One has 10 positive reviews out of 10, another 48 out of 50, and the third 186 out of 200. How should you trade off a higher rating against the confidence that comes from more observations?
This first video introduces a model and the binomial distribution. Later parts will cover Bayesian updating, continuous probability densities and beta distributions. Different assumptions and different goals can lead to different choices.
One preview is Laplace’s rule of succession: add one hypothetical positive and one hypothetical negative review, giving (positive + 1)/(total + 2). For the three sellers that yields 11/12, 49/52 and 187/202, so the second has the highest predicted chance of a positive next experience under that model.
Modeling a success rate
The true rate is unknown even when every observed review is positive.
The model treats each experience as positive or negative and assumes a constant but unknown success probability s for each seller. A perfect observed score need not imply s = 1. For example, if s = 0.95, the chance of ten independent positive experiences is 0.95^10, roughly 60%.
The broader question is uncertainty about the underlying long-run rate itself. A factory with two defective cars in an initial sample of 100 does not thereby know its true defect rate is exactly 2%; it must reason from limited data.
The binomial distribution
Simulation and a counting formula compute the probability of the data.
First ask the forward question: if we knew s, how likely would the observed review counts be? Simulating many groups of 50 reviews at s = 0.95 gives approximately 26.1% for exactly 48 positives.
The exact binomial formula is C(n, k) s^k (1−s)^(n−k). For 48 positives in 50 reviews, C(50,48) = 1,225 counts the possible placements of the two negatives; each particular arrangement has probability 0.95^48 × 0.05^2.
Multiplying those probabilities assumes the individual experiences are independent and share the same success probability. Under those assumptions the result is about 0.261, matching the simulation. A binomial distribution gives the probabilities of each possible count of successes in a fixed number of trials.
From likelihood toward inference
The next video will use Bayes’ rule to reason about the unknown rate.
The forward probability P(data | s) is distinct from the question we ultimately care about, P(s | data). To build intuition, vary s and plot how likely the observed 48 out of 50 is. The likelihood reaches its maximum at s = 0.96.
At s = 1, the observed two negative reviews would be impossible under the model, so the likelihood goes to zero. With many more reviews at the same observed proportion, the likelihood becomes more concentrated around 0.96.
A maximum-likelihood estimate alone does not answer the original prediction question. With 10 positive reviews out of 10, the likelihood s^10 is maximized at 1, but that does not justify certainty that the next experience will be positive.
A center-of-mass intuition points toward accounting for uncertainty across possible rates. The next step requires Bayes’ rule and probability densities; the presenter reserves that derivation for the next video. The earlier add-one rule corresponds to a particular prior and modeling assumptions.
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