Dependent and Independent Events

Dependent and Independent Events

Events are independent when one happening does not change the probability of the other; they are dependent when it does. Replacement is a powerful clue: drawing a card and replacing it resets the sample space, while drawing without replacement changes it. Use multiplication for an ‘and’ probability only after deciding whether the second probability is conditional.

Worked reasoning

From a bag with 3 red and 2 blue counters, probability of red then red without replacement is $3/5\times2/4=3/10$. The denominator changes because one counter is gone.

Exam method

1. State whether the first event changes the second sample space. 2. Write each probability in sequence. 3. Multiply for ‘and’ and simplify; explain replacement/no replacement.

Name the changing sample space

For every sequential probability, write a short note beside the second probability: ‘after one red removed’ or ‘replacement restores all counters’. That note explains the denominator and earns reasoning credit in a tree-free question. Distinguish mutually exclusive from independent: mutually exclusive events cannot happen together in one trial, whereas independent events can both happen but do not affect each other’s chances. The words ‘and’, ‘or’, ‘given’ and ‘without replacement’ are decision signals.

One-minute retrieval: Dependent and Independent Events

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.