An independent trial is a random experiment whose outcome probability is unaffected by outcomes of other trials. In a fair coin experiment, every flip has the same 0.5 probability of heads or tails, regardless of earlier flips. section 16.3
Use these features to recognize independent trials:
- Sample with replacement so each trial resets to the same available outcomes and probabilities. section 16.3
- Expect randomness in short sequences: streaks can occur without changing the probability of the next outcome. section 16.3
- Do not treat probabilities as guarantees; ten fair flips need not produce exactly five heads and five tails. section 16.3
- Bernoulli or binomial trials are independent trials with two possible outcomes. section 16.3
Coin flips and free throws
A fair coin can produce several consecutive heads or tails even when every flip remains independent. The same random pattern appears in NBA free-throw makes and misses: apparent streaks do not by themselves establish a hot hand or a causal improvement in shooting. section 16.3 section 16.6
sample(x = c(0, 1), size = 10, prob = c(0.5, 0.5), replace = TRUE) ## [1] 0 1 1 1 1 1 1 0 1 0
Run the R expression to simulate ten independent fair-coin trials, returning zeros and ones for the two outcomes. section 16.3
