Determine the maximum possible value that the probability of $B$ can assume. On a section of a lightly travelled country road, a maximum speed of 80 km/h is permitted. At one location on this section, the speed of passing cars is measured. In the following, only those journeys are considered where the drivers were able to choose their speed independently of one another. For the first 200 recorded journeys, the following distribution was obtained after classification into speed classes: [Figure] In 62\% of the 200 journeys, the driver was travelling alone, 65 of these solo drivers exceeded the speed limit. One journey is randomly selected from the 200 journeys. The following events are considered: $A$ : ``The driver was travelling alone.'' $S$ : ``The car was speeding.'' (1a) [5 marks] Show that events $A$ and $S$ are stochastically dependent, and give a possible reason for this in the context of the problem. The speed measurements are continued over a longer period. It turns out that the distribution of speeds measured to the nearest km/h can be approximately described by a binomial distribution with parameters $n = 100$ and $p = 0.8$. For example, $B ( 100 ; 0.8 ; 77 )$ approximately corresponds to the proportion of cars recorded at a speed of $77 \mathrm {~km} / \mathrm { h }$. (1b) [4 marks] Confirm by example for one of the two middle speed classes of the sample shown above that the determined number of journeys is consistent with the description by the binomial distribution. (1c) [2 marks] Using this binomial distribution, determine the smallest speed $v ^ { * }$ for which the following statement holds: ``In more than 95\% of the recorded journeys, $v ^ { * }$ is not exceeded.'' The police conduct a speed check at the measurement location. A speed of more than $83 \mathrm {~km} / \mathrm { h }$ constitutes a speeding violation. For simplicity, it should be assumed that the speed of a passing car is greater than $83 \mathrm {~km} / \mathrm { h }$ with a probability of 19\%. (2a) [4 marks] Calculate the number of speed measurements that must be performed at minimum so that with a probability of more than 99\% at least one speeding violation is recorded. (2b) [5 marks] If in a sample of 50 speed measurements the number of speeding violations is more than one standard deviation below the expected value, the police assume that there was effective warning of the speed check and abort the control. Determine the probability that the speed check is continued even though the probability of a speeding violation has dropped to 10\%.
Determine the maximum possible value that the probability of $B$ can assume.
On a section of a lightly travelled country road, a maximum speed of 80 km/h is permitted. At one location on this section, the speed of passing cars is measured. In the following, only those journeys are considered where the drivers were able to choose their speed independently of one another.
For the first 200 recorded journeys, the following distribution was obtained after classification into speed classes:\\
\textit{[Figure]}
In 62\% of the 200 journeys, the driver was travelling alone, 65 of these solo drivers exceeded the speed limit. One journey is randomly selected from the 200 journeys.\\
The following events are considered:\\
$A$ : ``The driver was travelling alone.''\\
$S$ : ``The car was speeding.''\\
\textbf{(1a)} [5 marks] Show that events $A$ and $S$ are stochastically dependent, and give a possible reason for this in the context of the problem.
The speed measurements are continued over a longer period. It turns out that the distribution of speeds measured to the nearest km/h can be approximately described by a binomial distribution with parameters $n = 100$ and $p = 0.8$. For example, $B ( 100 ; 0.8 ; 77 )$ approximately corresponds to the proportion of cars recorded at a speed of $77 \mathrm {~km} / \mathrm { h }$.
\textbf{(1b)} [4 marks] Confirm by example for one of the two middle speed classes of the sample shown above that the determined number of journeys is consistent with the description by the binomial distribution.
\textbf{(1c)} [2 marks] Using this binomial distribution, determine the smallest speed $v ^ { * }$ for which the following statement holds: ``In more than 95\% of the recorded journeys, $v ^ { * }$ is not exceeded.''
The police conduct a speed check at the measurement location.\\
A speed of more than $83 \mathrm {~km} / \mathrm { h }$ constitutes a speeding violation.\\
For simplicity, it should be assumed that the speed of a passing car is greater than $83 \mathrm {~km} / \mathrm { h }$ with a probability of 19\%.
\textbf{(2a)} [4 marks] Calculate the number of speed measurements that must be performed at minimum so that with a probability of more than 99\% at least one speeding violation is recorded.
\textbf{(2b)} [5 marks] If in a sample of 50 speed measurements the number of speeding violations is more than one standard deviation below the expected value, the police assume that there was effective warning of the speed check and abort the control. Determine the probability that the speed check is continued even though the probability of a speeding violation has dropped to 10\%.