Questions: Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S). Consider observing the direction for each of three successive vehicles. (Enter your answers in set notation. Enter EMPTY or ∅ for the empty set.) (a) List all outcomes in the event A that all three vehicles go in the same direction. A= (b) List all outcomes in the event B that all three vehicles take different directions. B= (c) List all outcomes in the event C that exactly two of the three vehicles turn right. C= (d) List all outcomes in the event D that exactly two vehicles go in the same direction. D=

Solution Steps

(a) To find the outcomes where all three vehicles go in the same direction, consider the possibilities where all are $R$, all are $L$, or all are $S$.

(b) For all three vehicles to take different directions, each vehicle must choose a unique direction from $R$, $L$, and $S$.

(c) To find outcomes where exactly two vehicles turn right, consider combinations where two vehicles are $R$ and the third is either $L$ or $S$.

To determine the outcomes where all three vehicles go in the same direction, consider the possibilities where each vehicle chooses the same direction from the set \(\{R, L, S\}\). The possible outcomes are:

• All vehicles turn right: \((R, R, R)\)
• All vehicles turn left: \((L, L, L)\)
• All vehicles go straight: \((S, S, S)\)

For all three vehicles to take different directions, each vehicle must choose a unique direction from \(\{R, L, S\}\). The permutations of these directions are:

• \((R, L, S)\)
• \((R, S, L)\)
• \((L, R, S)\)
• \((L, S, R)\)
• \((S, R, L)\)
• \((S, L, R)\)

To find outcomes where exactly two vehicles turn right, consider combinations where two vehicles are \(R\) and the third is either \(L\) or \(S\). The possible outcomes are:

• \((R, R, L)\)
• \((R, L, R)\)
• \((L, R, R)\)
• \((R, R, S)\)
• \((R, S, R)\)
• \((S, R, R)\)

Final Answer