Questions: There are 7 streets to be named after 7 tree types. Ash, Elm, Maple, Oak, Pine, Spruce, and Willow. A city planner randomly selects the street names from the list of 7 tree types. Compute the probability of each of the following events. Event A: The first three streets are Maple, Pine, and Ash, without regard to order. Event B: The first street is Maple, followed by Spruce and then Ash. Write your answers as fractions in simplest form.

Solution Steps

To solve this problem, we need to calculate the probabilities of two different events when naming streets after tree types.

For Event A, we need to find the probability that the first three streets are Maple, Pine, and Ash in any order. This involves calculating the number of favorable outcomes (all permutations of Maple, Pine, and Ash) and dividing by the total number of ways to arrange any three streets out of the seven available.

For Event B, we need to find the probability that the first street is Maple, the second is Spruce, and the third is Ash. This is a straightforward calculation of a single specific arrangement out of all possible arrangements of three streets.

To find the total number of ways to arrange 3 streets out of 7, we use the combination formula for selecting 3 streets from 7, multiplied by the factorial of 3 to account for the arrangements of those 3 streets: \[ \text{Total Ways} = \binom{7}{3} \times 3! = 35 \times 6 = 210 \]

Event A requires that the first three streets are Maple, Pine, and Ash in any order. The number of favorable outcomes (permutations of Maple, Pine, and Ash) is \(3! = 6\). Thus, the probability \(P(A)\) is given by: \[ P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{210} = \frac{1}{35} \]

Event B specifies that the first street is Maple, the second is Spruce, and the third is Ash. There is only one specific arrangement that satisfies this condition. Therefore, the probability \(P(B)\) is: \[ P(B) = \frac{1}{210} \]

Final Answer