Question

A parking lot has 7 unoccupied parking spaces in a row. One unoccupied parking space can only be occupied by one car. 4 cars enter the parking lot and occupy the 7 parking spaces at random. Find the probability that not all 4 cars are parked next to each other.

Answer 1

The probability that not all 4 cars are parked next to each other in a parking lot with 7 spaces is 31/35, which is approximately 0.8857.

Step-by-step explanation:

To find the probability that not all 4 cars are parked next to each other, we must consider two scenarios: the total number of ways 4 cars can park in 7 spaces, and the total number of ways that 4 cars can park adjacent to each other. We calculate the probability by subtracting the number of ways they can all park together from the total number of ways they can park, and then divide by the total number of ways.

There are C(7,4) ways to choose 4 spaces out of 7 for the cars. That's 7! / (4! (7-4)!) = 35 ways in total. To calculate the ways they can all park together, we need to think of the 4 cars as a block, which then has 4 possible positions (as moving this block from the first spot to the fourth spot). Therefore, there is 1 way to park the cars next to each other in each position, resulting in 4 ways.

To get the probability, we subtract the ways they can park together from the total ways and divide by the total ways:
Probability = (Total ways - Ways together) / Total ways = (35 - 4) / 35 = 31/35

The probability that not all 4 cars are parked next to each other is 31/35 or about 0.8857.

Answer 2

The probability that not all 4 cars are parked next to each other in a row of 7 parking spaces is 31/35 or approximately 0.8857.

Step-by-step explanation:

To calculate the probability that not all 4 cars are parked next to each other in a row of 7 parking spaces, we need to consider the total number of ways to park the cars and the number of unfavorable outcomes (where all 4 cars are parked next to each other).

Total ways to park the cars: There are 7 spaces and we need to choose 4 for the cars, which can be done in C(7,4) ways, using the combination formula which is C(n,k) = n! / (k! * (n - k)!). This gives us a total of 35 ways.

Ways to park all cars together: To have all 4 cars parked together, we can consider them as a block. There are 4 spaces that this block can be in (starting from space 1 to 4), so there are 4 ways this can happen.

Probability: The probability of the unfavorable outcome is the number of ways all cars are parked together divided by the total ways to park the cars. The probability that not all 4 cars are parked next to each other is the complement of this probability, which can be calculated by subtracting the probability of the unfavorable outcome from 1.

Therefore, P(not all together) = 1 - P(all together) = 1 - (4 / 35) = 31/35 or approximately 0.8857.