Question

A basket contains 12 fruits, of which 7 are not rotten. When one fruit is drawn at random, the probability that it is a rotten fruit is ____. (1) \(1 / 3\) (2) \(7 / 12\) (3) \(5 / 12\) (4) \(2 / 3\)

Short answer

Answer: (3) \(\frac{5}{12}\)

Step-by-step solution

Determine the number of rotten fruits in the basket

There are 12 fruits in total in the basket. 7 of them are not rotten. Therefore, the number of rotten fruits is: 12 (total fruits) - 7 (non-rotten fruits) = 5 (rotten fruits).

Calculate the probability of drawing a rotten fruit

Now that we know there are 5 rotten fruits in the basket, we can calculate the probability of drawing one rotten fruit out of the 12 total fruits: Probability = (Number of successful outcomes) / (Total number of outcomes) = (Number of rotten fruits) / (Total number of fruits) = \(\frac{5}{12}\).

Match the calculated probability with the given options

The calculated probability of drawing a rotten fruit is \(\frac{5}{12}\). Comparing this with the given options, we find that it matches option (3). Therefore, the correct answer is (3) \(\frac{5}{12}\).

Key concepts

Probability Calculation

Understanding probability calculation is essential when trying to determine the likelihood of events, such as drawing a certain type of fruit from a basket. In essence, probability quantifies how likely an event is to occur, and it is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 indicates certainty.

To calculate probability, use the formula:
\[ Probability = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} \]
In our example, the 'successful outcome' is drawing a rotten fruit. If we know the total number of fruits and the quantity that is rotten, we can apply this formula. Dividing the number of rotten fruits by the total fruits gives us the probability of drawing a rotten fruit.

It's crucial to ensure that the 'total number of outcomes' accounts for all possible occurrences under the condition of a random draw, and the 'number of successful outcomes' strictly refers to the event of interest, like the drawing of a rotten fruit in this case.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics and has applications in many scientific fields.

In the context of probability, combinatorics helps in calculating probabilities by counting the different ways certain outcomes can occur. For problems similar to the fruit-basket scenario, combinatorics isn’t directly used because we’re dealing with individual probabilities rather than combinations or permutations. However, combinatorics becomes vital when you have more complex situations, such as determining the probability of drawing multiple fruits in a particular order, or where certain restrictions are placed on the outcomes.

While the example given does not require combinatorial calculations, it's the foundation for understanding more advanced probability scenarios where the order of outcomes, or the grouping of outcomes, becomes significant.

Mathematical Problem Solving

Mathematical problem solving is a key skill that incorporates understanding the problem, devising a plan, carrying out the plan, and looking back to check the results. It’s a process that involves breaking down complex scenarios into manageable parts to find a solution.

In our fruit basket example, the problem was simplified into smaller steps. First, we needed to understand what constitutes a 'successful outcome', in this case, drawing a rotten fruit. Then, we created a plan to find out how many rotten fruits were there by subtracting the number of non-rotten fruits from the total. Executing this plan gave us the number of rotten fruits, which we then used to calculate the probability.

This systematic approach to solving mathematical problems allows for a structured analysis and solution pathway, which makes the process of learning and applying mathematical concepts more accessible.