Question

A shopping mall has four entrances, one on the North, one on the South, and two on the East. If you enter at random, shop and then exit at random, what is the probability that you enter and exit on the same side of the mall?

Short answer

The probability is \(\frac{1}{4}\).

Step-by-step solution

- Identify Total Possible Outcomes

Determine the total number of ways you can enter and exit the mall. Since there are 4 entrances, you can enter and exit in 4 x 4 = 16 different ways.

- Determine the Favorable Outcomes

Count the number of favorable outcomes where you enter and exit on the same side. There is 1 entrance each on the North and South sides, and 2 on the East side. Thus:- Enter and exit on North: 1 way- Enter and exit on South: 1 way- Enter and exit on East (both entrances): 2 ways.Total favorable outcomes = 1 (North) + 1 (South) + 2 (East) = 4.

- Calculate the Probability

The probability is the ratio of favorable outcomes to total possible outcomes. Therefore, the probability \[P = \frac{4}{16} = \frac{1}{4}.\]

Key concepts

Total Possible Outcomes

In probability, understanding the total possible outcomes is crucial. This means identifying all the unique ways a particular event can occur. In this exercise, you have four entrances to the mall—1 on the North, 1 on the South, and 2 on the East. You could enter and exit via any of these entrances. To find the total possible outcomes, you multiply the number of entrances by the number of exits, which is 4 x 4. Thus, you have 16 different ways you can enter and exit the mall. Each combination represents a unique outcome. So when we talk about 'total possible outcomes,' we’re referring to all these unique entry and exit combinations. Visualizing this can be helpful: imagine listing all pairs of entries and exits as (N, N), (N, S), (N, E1), (N, E2), and so on, until you have covered all 16 pairs.

Favorable Outcomes

Favorable outcomes are the specific events we are interested in when calculating probability. In this context, we want to know how many ways you can enter and exit on the same side of the mall. Let's break it down: you can enter and exit on the North side in 1 way, on the South side in 1 way, and on the East side in 2 ways—since there are 2 entrances/exists. Adding these up gives us: 1 (North) + 1 (South) + 2 (East) = 4 favorable outcomes. These are the outcomes where you end up on the same side after shopping. If you list them, they would look like: (N, N), (S, S), and (E1, E1), (E2, E2).

Probability Ratio

The probability ratio is the key to solving probability problems. It's the ratio of favorable outcomes to total possible outcomes. In our example, we have identified 4 favorable outcomes out of a total of 16 possible outcomes. The probability ratio therefore is: \[P = \frac{4}{16} = \frac{1}{4}\]. This fraction represents the chance of both entering and exiting the mall on the same side. Simplifying the fraction helps us understand that in simpler terms, the probability is 1 in 4.

Random Entry and Exit

Randomness means that each event (entry or exit) is equally likely. When you enter or exit the mall without any preference for a specific door, you are making a random choice. This randomness is a fundamental aspect of this exercise. Because every entrance or exit happens independently and randomly, each pair also has an equal probability of occurring. It's vital to understand this to grasp why the calculated probability applies uniformly across all scenarios. Random events ensure there's no bias, and every possible outcome has an equal chance of happening. This randomness is what underpins our probability calculations.