Question

A bowl contains ten chips numbered \(1,2, \ldots, 10\), respectively. Five chips are drawn at random, one at a time, and without replacement. What is the probability that two even-numbered chips are drawn and they occur on even- numbered draws?

Short answer

The probability that two even-numbered chips are drawn and they occur on even-numbered draws is \(\frac{5*5*4*4*5+5*5*4*5*4+5*4*5*4*4}{10*9*8*7*6}\)

Step-by-step solution

Determine the total number of draws

As five chips are drawn from ten, the total number of possible sequences is given by the permutation \(P(10,5)\), which equals \(10*9*8*7*6\).

Determine the number of successful draws

In order for two even-numbered chips to be drawn on even draws, they could either be drawn on the second and fourth draw, or the second and fifth, or the fourth and fifth. \n\nCase 1: EVEN chips are drawn on 2nd and 4th draws. This means the odd draws could be either even or odd numbered chips. The total possibilities for this case is \(5*5*4*4*5\).\n\nCase 2: EVEN chips are drawn on 2nd and 5th draws. The total possibilities for this case is also \(5*5*4*5*4\).\n\nCase 3: EVEN chips are drawn on 4th and 5th draws. The total possibilities for this case is \(5*4*5*4*4\).\n\nHence, the total number of successful possibilities is the sum of the three cases, which is \(5*5*4*4*5+5*5*4*5*4+5*4*5*4*4\)

Calculate the probability

The probability is the ratio between the number of successful draws and the total number of draws. By plugging in the numbers obtained in the previous steps, the probability is \(\frac{5*5*4*4*5+5*5*4*5*4+5*4*5*4*4}{10*9*8*7*6}\)

Key concepts

Permutations

In probability theory, the concept of permutations is essential for understanding how different arrangements or orders of a set can affect the outcome of a problem. A permutation refers to the arrangement of all or part of a set of objects in a specific order. It is different than a combination, which does not consider the order.
When drawing five chips from a set of ten in a specific sequence, as mentioned in the original exercise, we use the formula for permutations: \[P(n, r) = \frac{n!}{(n-r)!}\]Here, \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\). Therefore, for our problem with ten chips and five draws, the calculation becomes:

• \( n = 10 \)
• \( r = 5 \)
• \( P(10, 5) = \frac{10!}{(10-5)!} = 10 \times 9 \times 8 \times 7 \times 6 \)

This calculation provides us with the total number of possible sequences or ways to draw the chips.

Combinatorics

Combinatorics is a branch of mathematics concerning the counting, arrangement, and combination of objects. It's a crucial concept in solving probability problems involving different scenarios and configurations. By leveraging combinatorics, we can determine all possible ways in which specific conditions can be met.
In this exercise, we identify how the even-numbered chips can appear in specific positions during the drawing process. The arrangement must ensure two even chips are drawn specifically in even-numbered draws, for instance: - 2nd and 4th draws - 2nd and 5th draws - 4th and 5th draws For each of these cases, we calculate the number of successful scenarios. This process involves fixing the positions for the even-numbered chips and determining the combinations of remaining chips for the odd-numbered positions.
The calculated possible arrangements for these scenarios are then summed up to obtain the total number of successful draws.

Even and Odd Numbers

Understanding the distinction between even and odd numbers is fundamental, especially in this type of probability problem. Even numbers are integers divisible by 2 without a remainder, such as 2, 4, 6, 8, and 10. Odd numbers are not divisible by 2, such as 1, 3, 5, 7, and 9.
In the context of the exercise, the chips numbered from 1 to 10 need to be categorized based on whether they are even or odd in order to set up the scenarios properly. This categorization dictates which numbers can be selected for certain positions in the drawing, ensuring that the probabilities calculated are accurate.
In scenarios like drawing even-numbered chips specifically on even-numbered draws, understanding this classification allows for correctly setting parameters that match the desired outcome. This helps calculate probabilities of certain configurations occurring, reinforcing the importance of even and odd number properties.