A certain five-sided die is biased with faces numbered 0 to 4 . The score, Y , on each throw is a random variable with probability distribution given by:
\(Y\)
0
1
2
3
4
\(\mathrm { P } ( Y = y )\)
\(a\)
\(b\)
\(c\)
0.1
0.15
where \(a\), \(b\) and \(c\) are constants.
$$\begin{aligned}
& \mathrm { P } ( Y = 1 ) = \mathrm { P } ( Y \geq 3 )
& \mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 ) - 0.1
\end{aligned}$$
Find the values of \(a , b\) and \(c\). [0pt]
[4 marks]
The same die is thrown 10 times. Find the probability that there are not more than 4 throws on which the score is 3 , stating the distribution used as well as any modelling assumptions made. [0pt]
[4 marks]
A game uses the same biased die. The die is thrown once. If it shows 1, 3 or 4 then this number is the final score. If it shows 0 or 2 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
(a) Find the probability that the final score is 3 .
(b) Given that the die is thrown twice, find the probability that the final score is 3 . [0pt]
[3 marks] [0pt]
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