Independence, Trees and Streaks: Quick Revision

A brief note on Independence, Trees and Streaks. Read it before a test. Try each question at the end before you open its answer.

Independent events

• Meaning: one event does not change the chance of the other.
• Multiplication rule: $P(A \cap B) = P(A) \times P(B)$ for independent events.
• At least one: $P(\text{at least one}) = 1 - P(\text{none})$.
• Not independent: weather on two days, two cards from one pack, two facts about one die.

Trees and replacement

• Tree: branches from a point add up to $1$; multiply along a path; add the paths.
• Without replacement: the second branches use the counts that are left.
• Big population: draws without replacement are almost independent.

Streaks

• Gambler's fallacy: after a streak, the next independent result is not "due".
• Every sequence: $n$ fair tosses give each particular sequence the probability $\left(\dfrac{1}{2}\right)^n$.
• Long runs are normal: in $100$ fair tosses, a run of $5$ or more equal faces is very likely.

A worked example

Two made-up machines work independently, with probabilities $0.8$ and $0.6$.

1. Both work. $0.8 \times 0.6 = 0.48$.
2. At least one works. Both fail: $0.2 \times 0.4 = 0.08$. So $1 - 0.08 = 0.92$.
3. Exactly one works. $0.92 - 0.48 = 0.44$.

Practice with new numbers

A program to try

The program checks the worked example, and then finds the chance that two cards from one pack are both red.

Mistakes to avoid

• Multiplying linked events. Check independence first.
• Adding along a path. Multiply along it.
• Thinking a result is due. Independent tries have no memory.
• Forgetting an order. "Exactly one" often has two paths.

Check yourself

1. Two independent events have probabilities $0.3$ and $0.6$. Find the probability that both happen.

2. A bag holds $3$ blue and $5$ green counters. Two are taken without replacement. Find the probability that both are blue.

3. A fair die is rolled $3$ times. Find the probability of at least one six.

4. After $4$ heads in a row, what is the chance that a fair coin lands tails?

Sources

• OpenStax, Introductory Statistics, Chapter 3, the sections on independent events and on tree diagrams.