Two way tables and tree diagrams Probability 12C Two way tables and tree diagrams

Two way tables When more than one event has to be considered, a visual representation of the sample space is helpful in calculating the probabilities of various events. Two-way tables A two-way table (sometimes referred to as a lattice diagram) is able to represent two events in a table form, when the two events occur at the same time A two-way table for the experiment of tossing a coin and rolling a die simultaneously is shown in the following table. Possibilities for dice roll Possibilities for coin toss All possible outcomes/ combinations

Worked example Two dice are rolled. The outcome is the pair of numbers shown. (a) Show the results on a two-way table. Dice 2 possibilities 1 2 3 4 5 6 Dice 1 possibilities 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Dice 2 possibilities 1 2 3 4 5 6 Dice 1 possibilities

How many total possibilities are there? 6 x 6 = 36 total possibilities (b) Calculate the probability of obtaining an identical ordered pair; that is, Pr[(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)] How many total possibilities are there? 6 x 6 = 36 total possibilities How many ordered pairs are there? 6 Therefore, Pr(ordered pair) = 6 36 = 1 6 Dice 2 possibilities 1 2 3 4 5 6 Dice 1 possibilities 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6

Tree diagrams Tree diagrams are very helpful when there are multiple events; for example, when a coin is tossed twice. These are events that happen one after the other, not at the same time (like two way tables) Each stage of a multiple event experiment produces a part of a tree. For example, this tree diagram shows the outcomes of a coin being tossed once, then tossed again To work put the probability of these events occurring, multiply them together, e.g. Pr(HT) = 0.5 x 0.5 = 0.25 When you add all the probabilities together, they should add to one

Worked example Three coins are tossed simultaneously. Draw a tree diagram for the experiment. Calculate the following probabilities. a Pr(3 Heads) b Pr(2 Heads) c Pr(at least 1 Head)

Three coins are tossed simultaneously Three coins are tossed simultaneously. Draw a tree diagram for the experiment. Coin 1 Coin 2 Coin 3 H T 1 2 HHH HHT HTH HTT THH THT TTH TTT 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

Calculate the following probabilities. a Pr(3 Heads) b Pr(2 Heads) c Pr(at least 1 Head) (a) Pr(HHH) = 1 2 x 1 2 x 1 2 = 1 8 (b) Pr(two H) = Pr(HHT) + Pr(HTH) + Pr(THH) = 1 8 + 1 8 + 1 8 = 3 8 (c) Pr(at least one H) = How many outcomes have at least one H? All except TTT Pr(TTT) = 1 8 1 – Pr(TTT) = 1 - 1 8 = 7 8

Worked example The letters A, B, C and D are written on identical pieces of card and placed in a box. A letter is drawn at random from the box. Without replacing the first card, a second one is drawn. Use a tree diagram to find: a Pr(first letter is A) b Pr(second letter is B) c Pr(both letters are the same).

AB AC AD BA BC BD CA CB CD DA DB DC Three coins are tossed simultaneously. Draw a tree diagram for the experiment. First draw Second draw 1 3 AB AC AD BA BC BD CA CB CD DA DB DC Pr(AB) = 1 4 x 1 3 = 1 12 Pr(AC) = 1 4 x 1 3 = 1 12 Pr(AD) = 1 4 x 1 3 = 1 12 Pr(BA) = 1 4 x 1 3 = 1 12 Pr(BC) = 1 4 x 1 3 = 1 12 Pr(BD) = 1 4 x 1 3 = 1 12 Pr(CA) = 1 4 x 1 3 = 1 12 Pr(CB) = 1 4 x 1 3 = 1 12 Pr(CD) = 1 4 x 1 3 = 1 12 Pr(DA) = 1 4 x 1 3 = 1 12 Pr(DB) = 1 4 x 1 3 = 1 12 Pr(DC) = 1 4 x 1 3 = 1 12 A B C D 1 3 1 3 Remember, the card is removed after the first draw, leaving the 3 leftover cards in the second draw 1 4 1 3 1 3 1 4 1 3 1 3 1 4 1 3 1 3 1 4 1 3 1 3 1 3

Pr(first letter is A) = Pr(AA) + Pr(AB) + Pr(AC) + Pr(AD) = 1 12 + 1 12 + 1 12 = 3 12 = 1 4 (b) Pr(second letter B) = Pr(AB) + Pr(CB) + Pr(DB) (c) Pr(both letters the same) = Pr(AA) + Pr(BB) + Pr(CC) + Pr(DD) = 0

Questions to do Exercise 12C p410: 1, 3, 7, 8, 10, 11, 12