A student must go to his school each morning by 8:00 a.m. He takes the bicycle 7 days out of 10 and the bus the rest of the time. On days when he takes the bicycle, he arrives on time in $99.4\%$ of cases and when he takes the bus, he arrives late in $5\%$ of cases. A date is chosen at random during the school period and we denote by $V$ the event ``The student goes to school by bicycle'', $B$ the event ``the student goes to school by bus'' and $R$ the event ``The student arrives late at school''.

1. Translate the situation using a probability tree.
2. Determine the probability of the event $V \cap R$.
3. Prove that the probability of the event $R$ is 0.0192
4. On a given day, the student arrived late at school. What is the probability that he went there by bus?

A light bulb manufacturer has two machines, denoted A and B. Machine A provides $65\%$ of production, and machine B provides the rest. Some light bulbs have a manufacturing defect:

• at the output of machine $\mathrm{A}$, $8\%$ of light bulbs have a defect;
• at the output of machine B, $5\%$ of light bulbs have a defect.

The following events are defined:

• A: ``the light bulb comes from machine A'';
• B: ``the light bulb comes from machine B'';
• $D$: ``the light bulb has a defect''.

1. A light bulb is randomly selected from the total production of one day. a. Construct a probability tree representing the situation. b. Show that the probability of drawing a light bulb without a defect is equal to 0.9305. c. The light bulb drawn has no defect. Calculate the probability that it comes from machine A.
2. 10 light bulbs are randomly selected from the production of one day at the output of machine A. The size of the stock allows us to consider the trials as independent and to assimilate the draws to draws with replacement. Calculate the probability of obtaining at least 9 light bulbs without a defect.

The operator of a communal forest decides to fell trees in order to sell them, either to residents or to businesses. It is assumed that:

• among the felled trees, $30 \%$ are oaks, $50 \%$ are firs and the others are trees of secondary species (which means they are of lesser value);
• $45.9 \%$ of the oaks and $80 \%$ of the firs felled are sold to residents of the commune;
• three quarters of the felled trees of secondary species are sold to businesses.

Among the felled trees, one is chosen at random. The following events are considered:

• C: ``the felled tree is an oak'';
• $S$: ``the felled tree is a fir'';
• $E$: ``the felled tree is a tree of secondary species'';
• $H$: ``the felled tree is sold to a resident of the commune''.

1. Construct a complete weighted tree representing the situation.
2. Calculate the probability that the felled tree is an oak sold to a resident of the commune.
3. Justify that the probability that the felled tree is sold to a resident of the commune is equal to 0.5877.
4. What is the probability that a felled tree sold to a resident of the commune is a fir? The result will be given rounded to $10^{-3}$.