We denote by $p$ the unknown proportion of young people aged 16 to 24 years who practice illegal downloading on the internet at least once a week.

A young person participating in protocol $( \mathscr { P } )$ is randomly selected. The protocol $( \mathscr { P } )$ is as follows: each young person rolls a fair 6-sided die; if the result is even, the young person answers sincerely; if the result is ``1'', the young person must answer ``Yes''; if the result is ``3 or 5'', the young person must answer ``No''.

We denote: $R$ the event ``the result of the roll is even'', $O$ the event ``the young person answered Yes''.

\section*{1. Probability calculations}
Reproduce and complete the weighted tree diagram.

Deduce that the probability $q$ of the event ``the young person answered Yes'' is:
$$q = \frac { 1 } { 2 } p + \frac { 1 } { 6 }$$

\section*{2. Confidence interval}
a. At the request of Hadopi, a polling institute conducts a survey according to protocol $( \mathscr { P } )$. On a sample of size 1500, it counts 625 ``Yes'' responses.\\
Give a confidence interval, at the 95\% confidence level, for the proportion $q$ of young people who answer ``Yes'' to such a survey, among the population of young French people aged 16 to 24 years.\\
b. What can be concluded about the proportion $p$ of young people who practice illegal downloading on the internet at least once a week?