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''.
Translate the situation using a probability tree.
Determine the probability of the event $V \cap R$.
Prove that the probability of the event $R$ is 0.0192
On a given day, the student arrived late at school. What is the probability that he went there by bus?
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''.
\begin{enumerate}
\item Translate the situation using a probability tree.
\item Determine the probability of the event $V \cap R$.
\item Prove that the probability of the event $R$ is 0.0192
\item On a given day, the student arrived late at school. What is the probability that he went there by bus?
\end{enumerate}