It is now assumed that, in the store:

• $80\%$ of the padlocks offered for sale are budget models, the others are high-end;
• $3\%$ of high-end padlocks are defective;
• $7\%$ of padlocks are defective.

A padlock is randomly selected from the store. We denote:

• $p$ the probability that a budget padlock is defective;
• $H$ the event: ``the selected padlock is high-end'';
• $D$ the event: ``the selected padlock is defective''.

1. Represent the situation using a probability tree.
2. Express $P(D)$ as a function of $p$. Deduce the value of the real number $p$.

Is the result obtained consistent with that of question A-2?
3. The selected padlock is in good condition. Determine the probability that it is a high-end padlock.

We consider a group from the population of another country subjected to the same test with sensitivity 0.8 and specificity 0.99.
In this group the proportion of individuals with a positive test is 29.44\%.
An individual is chosen at random from this group; what is the probability that they have been infected?

A survey of 320 students at a school regarding membership in the mathematics club found that 60\% of male students and 50\% of female students joined the mathematics club. Let $p _ { 1 }$ be the probability that a randomly selected student from those who joined the mathematics club is male, and let $p _ { 2 }$ be the probability that a randomly selected student from those who joined the mathematics club is female. When $p _ { 1 } = 2 p _ { 2 }$, what is the number of male students at this school? [4 points]
(1) 170
(2) 180
(3) 190
(4) 200
(5) 210