Question
A biased four-sided die, A, is rolled. Let X be the score obtained when die A is rolled. The
probability distribution for X is given in the following table.
(a) Find the value of p . [2]
(b) Hence, find the value of E (X ) . [2]
A second biased four-sided die, B, is rolled. Let Y be the score obtained when die B is rolled.
The probability distribution for Y is given in the following table.
(c) (i) State the range of possible values of r .
(ii) Hence, find the range of possible values of q . [3]
(d) Hence, find the range of possible values for E (Y ) . [3]
Agnes and Barbara play a game using these dice. Agnes rolls die A once and Barbara rolls die B once. The probability that Agnes’ score is less than Barbara’s score is \(\frac{1}{2}\)
(e) Find the value of E (Y ) . [6]
Answer/Explanation
Ans
Question
The probability distribution of a discrete random variable X is given by \[{\rm{P}}(X = x) = \frac{{{x^2}}}{{14}}{\text{, }}x \in \left\{ {1{\text{, }}2{\text{, }}k} \right\}{\text{, where}} k > 0\] .
a.Write down \({\rm{P}}(X = 2)\) .[1]
b.Show that \(k = 3\) .[4]
c.Find \({\rm{E}}(X)\) .[2]
Answer/Explanation
Markscheme
\({\rm{P}}(X = 2) = \frac{4}{{14}}\) \(\left( { = \frac{2}{7}} \right)\) A1 N1
[1 mark]
\({\rm{P}}(X = 1) = \frac{1}{{14}}\) (A1)
\({\rm{P}}(X = k) = \frac{{{k^2}}}{{14}}\) (A1)
setting the sum of probabilities \( = 1\) M1
e.g. \(\frac{1}{{14}} + \frac{4}{{14}} + \frac{{{k^2}}}{{14}} = 1\) , \(5 + {k^2} = 14\)
\({k^2} = 9\) (accept \(\frac{{{k^2}}}{{14}} = \frac{9}{{14}}\) ) A1
\(k = 3\) AG N0
[4 marks]
correct substitution into \({\rm{E}}(X) = \sum {x{\rm{P}}(X = x)} \) A1
e.g. \(1\left( {\frac{1}{{14}}} \right) + 2\left( {\frac{4}{{14}}} \right) + 3\left( {\frac{9}{{14}}} \right)\)
\({\rm{E}}(X) = \frac{{36}}{{14}}\) \(\left( { = \frac{{18}}{7}} \right)\) A1 N1
[2 marks]
Question
The random variable X has the following probability distribution.
Given that \({\rm{E}}(X) = 1.7\) , find q .
Answer/Explanation
Markscheme
correct substitution into \({\rm{E}}(X) = \sum {px} \) (seen anywhere) A1
e.g. \(1s + 2 \times 0.3 + 3q = 1.7\) , \(s + 3q = 1.1\)
recognizing \(\sum {p = 1} \) (seen anywhere) (M1)
correct substitution into \(\sum {p = 1} \) A1
e.g. \(s + 0.3 + q = 1\)
attempt to solve simultaneous equations (M1)
correct working (A1)
e.g. \(0.3 + 2q = 0.7\) , \(2s = 1\)
\(q = 0.2\) A1 N4
[6 marks]
Question
[without GDC]
Each of the following 10 words is placed on a card and put in a hat.
ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN
We pick a card at random. Let \(X\) be the size (number of letters) of the corresponding word.
(a) Give the probability distribution for \(X\) (i.e. the table of probabilities)
(b) Find the expected number of \(X\).
Answer/Explanation
Ans
(a) 
(b) Find \(E(X) = 3.9\)