Question 6 - A-Level Maths

Exam Board	Edexcel
Module	S1 (Statistics 1)
Year	2017
Session	October
Topic	Discrete Probability Distributions
Type	Construct probability distribution from scenario

The discrete random variable \(D\) with the following probability distribution represents the score when a 4-sided die is rolled.

\(d\)	1	2	3	4
\(\mathrm { P } ( D = d )\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)

Write down the name of this distribution. The die is used to play a game and the random variable \(X\) represents the number of points scored. The die is rolled once and if \(D = 2,3\) or 4 then \(X = D\). If \(D = 1\) the die is rolled a second time and \(X = 0\) if \(D = 1\) again, otherwise \(X\) is the sum of the two scores on the die.
Show that the probability of scoring 3 points in this game is \(\frac { 5 } { 16 }\)
Find the probability of scoring 0 in this game. The table below shows the probability distribution for the remaining values of \(X\).
\(x\) 0 2 3 4 5
\(\mathrm { P } ( X = x )\) \(\frac { 1 } { 4 }\) \(\frac { 5 } { 16 }\) \(\frac { 1 } { 16 }\)
Find \(\mathrm { E } ( X )\)
Find \(\operatorname { Var } ( X )\) The discrete random variable \(R\) represents the number of times the die is rolled in the game.
Write down the probability distribution of \(R\). The random variable \(Y = 2 R + 0.5\)
Show that \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\) The game is played once.
Find \(\mathrm { P } ( X > Y )\)

\(x\)	0	2	3	4	5
\(\mathrm { P } ( X = x )\)		\(\frac { 1 } { 4 }\)		\(\frac { 5 } { 16 }\)	\(\frac { 1 } { 16 }\)