A certain university has 10 vehicles available for use by faculty and staff. Six of these are vans and four are cars. On a particular day, only two requests for vehicles have been made. Suppose that the two vehicles to be assigned are chosen in a completely random fashion from among the 10 . a. Let $E$ denote the event that the first vehicle assigned is a van. What is $P(E) ?$ b. Let $F$ denote the event that the second vehicle assigned is a van. What is $P(F \mid E)$ ? c. Use the results of Parts (a) and (b) to calculate $P(E$ and $F)$ (Hint: Use the definition of $P(F \mid E)$.)
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A certain university has 10 vehicles available for use by faculty and staff. Six of these are vans and four are cars. On a particular day, only two requests for vehicles have been made. Suppose that the two vehicles to be assigned are chosen in a completely random fashion from among the 10 . a. Let $E$ denote the event that the first vehicle assigned is a van. What is $P(E)$ ? b. Let $F$ denote the probability that the second vehicle assigned is a van. What is $P(F \mid E)$ ? c. Use the results of Parts (a) and (b) to calculate $P(E$ and $F)$ (Hint: Use the definition of $P(F \mid E) .$ )
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A certain university has 19 vehicles available for use by faculty and staff. Six of these are vans and 13 are cars. On a particular day, only two requests for vehicles have been made. Suppose that the two vehicles to be assigned are chosen in a completely random fashion from among the 19. (Enter your answers as fractions.) (a) Let E denote the event that the first vehicle assigned is a van. What is the value of P(E)? (b) Let F denote the probability that the second vehicle assigned is a van. What is the value of P(F | E)? (c) Use the results of parts (a) and (b) to calculate P(E ∩ F).
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A school has 250 employees categorized by task and gender in the following table. $$\begin{array}{|l|c|c|c|}\hline & \text { Teaching } & \text { Administrative } & \text { Support } \\\hline \text { Male } & 84 & 14 & 52 \\\hline \text { Female } & 56 & 26 & 18 \\\hline\end{array}$$ An employee is randomly selected. Let $A$ be the event that he/she is an administrative staff member, T teaching staff, S support, $M$ male, and F female. a) Write down the following probabilities: $P(F), P(F \cap T), P\left(F \cup A^{\prime}\right), P\left(F^{\prime} | A\right)$ b) Which events are independent of $F$, which are mutually exclusive to $F$. Justify your choices. c) (i) Given that $90 \%$ of teachers, as well as $80 \%$ of the administrative staff and $30 \%$ of the support staff, own cars, find the probability that a staff member chosen at random owns a car. (ii) Knowing that the randomly chosen staff member owns a car, find the probability that he/she is a teacher.
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