Calculate the probability of independent and dependent combined events,…

The idea

For sequential events, multiply probabilities along each tree path, then add non-overlapping paths when any satisfies the question. The general rule is P(A and B) = P(A) × P(B given A). If A and B are independent, knowing A happened does not change B. For two independent flips of a biased coin with P(Head) = 0.6, the second-flip branches repeat the first and P(two Heads) = 0.6 × 0.6 = 0.36. For dependent events, branch probabilities change. From a bag with 5 red and 3 blue counters, without replacement, P(Red first) = 5/8 and P(Red second given Red first) = 4/7, so P(two Reds) = 5/8 × 4/7 = 20/56. A sample-space diagram or two-way table can be clearer for simultaneous events such as two dice. Treat its outcomes as equally likely only when the dice are fair. State the assumptions used: random selection, replacement, equally likely outcomes and justified independence.

What earns the marks

• Correctly calculating probabilities for the first event or set of branches.
• For dependent events (without replacement), correctly adjusting probabilities on the second branches: reducing the denominator by one, and reducing the numerator only for the same outcome.
• Multiplying probabilities along a branch to find the probability of a specific combined outcome.
• Adding the probabilities of all relevant paths that satisfy the question's condition (e.g., 'probability of getting one of each colour').
• Correctly constructing and using a sample space diagram or two-way table to find a required probability.

Common mistakes and how to put them right

• In 'without replacement' problems, keeping the denominator the same for the second event.

  Put it right: The total number of items has decreased by one, so the denominator must also decrease by one for the second event.

• Adding probabilities along the branches of a tree diagram instead of multiplying.

  Put it right: The 'AND' rule for a sequence of events means you must multiply along the path. The 'OR' rule means you add the probabilities of separate, successful paths.

• Forgetting to consider all valid outcomes, e.g., for P(one of each colour), calculating P(Red then Blue) but not P(Blue then Red).

  Put it right: If the order is not specified, you must find the probability of all possible orderings that satisfy the condition and add them together.

Exam tips

• Before starting, find the words 'with replacement' or 'without replacement'. This is the key to deciding if the probabilities on the second set of branches change or stay the same.
• For questions asking for the probability of 'at least one' of something, it is almost always quicker to calculate 1 – P(none of them) instead.
• Choose the best representation. Tree diagrams are excellent for sequential events. Sample space diagrams (or two-way tables) are often clearer for simultaneous events like rolling two dice.