The Probability of Selecting Two Different Kinds of Fruit from a Fridge

In this article, we will delve into a fascinating math problem: calculating the probability of selecting two different kinds of fruit when picking sequentially from a fridge containing an equal number of various fruits. This problem is always intriguing and can be solved using basic combinatorics.

Problem Statement

A fridge contains 24 fruits in total, equally comprising 8 oranges, 8 apples, and 8 pears. We randomly select two fruits one after another without replacing the first fruit. What is the probability that the two fruits selected are of different kinds?

Approach 1: Using Combinatorics

To solve this problem, let's first understand the number of ways to select 2 fruits out of 24. This is a combination problem, and the number of ways to choose 2 fruits from 24 is given by the combination formula:

[ binom{24}{2} frac{24!}{2!(24-2)!} frac{24 times 23}{2 times 1} 276 ]

Now, let's find the number of favorable outcomes, i.e., the number of ways to select two different kinds of fruits.

Step 1: First Fruit Selection

The first fruit can be any of the 24 available fruits. Therefore, there are 24 ways to choose the first fruit.

Step 2: Second Fruit Selection

For the second fruit to be of a different kind, we need to consider the remaining 16 fruits (since there are 8 of each kind, and we have already chosen one). There are 23 remaining fruits in total (since we don't replace the first fruit).

Therefore, the number of ways to choose the second fruit from the remaining 23 fruits, ensuring it is of a different kind, is 16. This is because there are 16 fruits of the different kinds (8 apples and 8 pears if the first fruit was an orange, and so on).

The number of favorable outcomes is thus calculated as:

[ 24 times 16 384 ]

Approach 2: Using Conditional Probability

Another way to solve this problem is by using the concept of conditional probability. Let's break it down:

Step 1: Probability of First Fruit Being of Any Kind

First, we select any fruit from the 24 available. Since every fruit is equally likely, the probability of this event is:

[ frac{1}{1} 1 ]

This might seem trivial, but it's a good starting point.

Step 2: Probability of the Second Fruit Being of a Different Kind

For the second fruit to be of a different kind, we need to ensure that the second pick does not match the first pick. There are 23 fruits left, and out of these, 16 are of a different kind from the first fruit. Therefore, the probability of picking a different kind of fruit as the second pick is:

[ frac{16}{23} ]

The combined probability of these events happening in sequence is the product of their individual probabilities:

[ 1 times frac{16}{23} frac{16}{23} ]

So, the probability of selecting two different kinds of fruits in a row is: