A probability distribution maps every possible value of a random variable to its probability. Think of the variable as the result of a chance-driven procedure: adult height, raindrops in a minute, a die roll, or whether a water-damaged laptop recovers. Taken together, the probabilities must total 1.0 (100%). section 3.4 section 3.1
Distributions make uncertainty usable. Rather than demand one certain answer, they describe all plausible outcomes and how likely each is. Probabilities may come from observed data, personal or expert belief, or a mix; that origin matters because biased or incomplete information can make a distribution less trustworthy. section 3.0 section 3.3
Table 3.1 Probability distributions covered in this chapter
| Probability distribution |
Discrete or continuous? |
Description |
Example |
|---|---|---|---|
| Bernoulli |
Discrete |
Describes a single random experiment with exactly two outcomes: success or failure |
After a spill, there’s a 70% chance my MacBook Air will recover and a 30% chance it will not. |
| Discrete (General) |
Discrete |
Assigns probabilities to a finite number of outcomes for an event |
Probabilities of selling different colors of a video game controller |
| Binomial |
Discrete |
Predicts the probability of so many successes out of so many trials, assuming a fixed probability for each success |
Probability of seeing 7 heads in flipping a 50/50 coin 10 times |
| Beta |
Continuous |
Shows the likelihood of different success rates for an event to occur, given so many successes |
For a possibly rigged coin, shows the most possible outcomes of heads on a single flip, given that 8 heads occurred out |
| Poisson |
Discrete |
Measures the probability of how many times an event is likely to occur for a given interval |
Predicted number of server requests in a given minute |
| Exponential |
Continuous |
Predicts the time or |
Predicted time between server requests |
| Normal |
Continuous |
Predicts a continuous variable as a symmetrical bell curve defined by a mean and standard deviation |
The body temperature of any given adult |
Compare distribution families by whether outcomes are discrete or continuous, the problem cues they fit, and a concrete use case. section 3.4
Choose a distribution by first identifying what kind of outcome the problem describes:
- A single two-outcome experiment calls for a Bernoulli distribution: success or failure, yes or no. section 3.4 section 3.5
- A finite set of possible outcomes can use a general discrete distribution that assigns each outcome a probability. section 3.4
- Repeated trials, elapsed time, and observed successes or failures point toward different specialized distributions. section 3.4
A 70% recovery chance
Let X represent laptop recovery after a spill. Assign X = 1 to recovery and X = 0 to non-recovery. A Bernoulli distribution can set P(X=1)=0.70 and P(X=0)=0.30. The values add to 1.0, and the distribution can be shown as two bars, stored in a Python dictionary, or sampled with numpy for a simulation. section 3.5 section 3.1
A distribution is not a promise about what will happen once. It is a structured model of uncertainty. Small samples can give a misleading impression of consistency or spread, so interpret patterns alongside sample size and the limits of the data. section 2.20 section 3.0
