4MA0 Higher June 2013 Paper 3R Q18

EdexcelHigherOld spec6 marksTree Diagrams

18 Boris and Nigel play games of chess against each other in a match.
In each game, Boris wins or Nigel wins or the game is a draw.

When a player wins a game, he wins the match.
When a game is a draw, the players play another game against each other.
Boris and Nigel play a maximum of 3 games.

The probability that Boris wins a game is \(\dfrac{1}{3}\)

The probability that a game is a draw is \(\dfrac{1}{2}\)

(a) Complete the probability tree diagram.

Incomplete probability tree over three games; each game branches to Boris wins, Draw and Nigel wins, with the next game following Draw; first game shows 1/3 for Boris wins and 1/2 for Draw

(3)

(b) Calculate the probability that Boris wins the match. (3)

Mark scheme (a)

Scheme	Marks
\(1 - \dfrac{1}{2} - \dfrac{1}{3}\left(= \dfrac{1}{6}\right)\)	M1
	A1
correct fractions on branches	A1
	(3)

Notes

A1: for \(\dfrac{1}{6}\) oe

A1: correct values in correct places on full tree

Note: (simplest form of fractions is not necessary)
(accept \(\dfrac{1}{6}\) and/or \(\dfrac{1}{3}\) rounded or truncated to 2 or more decimal places eg 0.16 , 0.17 , 0.33 etc)

SC : If M1 cannot be awarded then award B1 if top two branches in 2nd and 3rd games are correct

Mark scheme (b)

Scheme	Marks
\(\dfrac{1}{3} + \dfrac{1}{2} \times \text{“}\tfrac{1}{3}\text{”} + \dfrac{1}{2} \times \text{“}\tfrac{1}{2}\text{”} \times \text{“}\tfrac{1}{3}\text{”}\)	M2
\(\dfrac{7}{12}\)	A1
	(3)

Notes

M2: M1 for \(\dfrac{1}{2} \times \text{“}\tfrac{1}{3}\text{”}\) or \(\dfrac{1}{2} \times \text{“}\tfrac{1}{2}\text{”} \times \text{“}\tfrac{1}{3}\text{”}\)

A1: accept 0.583... rounded or truncated to 2 or more sf

Alternative method for (b)

Scheme	Marks
\(1 - \left(\text{“}\tfrac{1}{6}\text{”} + \dfrac{1}{2} \times \text{“}\tfrac{1}{6}\text{”} + \dfrac{1}{2} \times \text{“}\tfrac{1}{2}\text{”} \times \text{“}\tfrac{1}{6}\text{”} + \dfrac{1}{2} \times \text{“}\tfrac{1}{2}\text{”} \times \text{“}\tfrac{1}{2}\text{”}\right)\)	M2
\(\dfrac{7}{12}\)	A1

Notes

M2: M1 for \(\dfrac{1}{2} \times \text{“}\tfrac{1}{6}\text{”}\) or \(\dfrac{1}{2} \times \text{“}\tfrac{1}{2}\text{”} \times \text{“}\tfrac{1}{6}\text{”}\) or \(\dfrac{1}{2} \times \text{“}\tfrac{1}{2}\text{”} \times \text{“}\tfrac{1}{2}\text{”}\)

A1: accept 0.583... rounded or truncated to 2 or more sf