WJEC Maths for A2 – Applied

1.2 The conditional probability formula 11 Conditional probabilities can be shown using the following tree diagram: A A Ԣ P( A ) P( A Ԣ ) P( B | A ) P( B Ԣ | A ) P( B | A Ԣ ) P( B Ԣ | A Ԣ ) B B Ԣ B B Ԣ The probability of A and B both occurring is given by This is an important law and is called the multiplication law for dependent events. By using the above tree diagram, we have the following results: P( A ∩ B′  ) = P( A ) × P( B′  | A ) P( A′ ∩ B ) = P( A’  ) × P( B  | A′  ) P( A′ ∩ B′  ) = P( A′  ) × P( B′  | A′  ) Notice the way any of these formulae can be produced using the formula given in the formula booklet (i.e. P( A ∩ B ) = P( A ) × P( B | A ) simply by changing A to A′ or B  to  B′ or both A and B to A′ and B′  ). We can also have P( A ∩ B ) = P( B ) × P( A | B ) 1.2 The conditional probability formula The multiplication law for dependent events is and also, Examples 1 Two events A and B occur such that P( A′  ) = 0.6, P( B  | A ) = 0.7, P( B  | A′  ) = 0.4 . (a) Copy and complete the tree diagram shown below to illustrate the above information. A A Ԣ 0.6 B B Ԣ B B Ԣ Notice from the tree diagram that P( B | A ) + P( B ′| A ) = 1 and P( B | A ′) + P( B ′| A ′) = 1 as the total probability for a branch has to equal one. This formula is included in the formula booklet and need not be remembered. These equations can be found by following the various sets of branches for the above tree diagram. This is not included in the formula booklet. Notice the way the A and B are swapped around on the right-hand side of the formula.