Question

A bag contains 8 balls numbered 1 to 8 . If 2 balls are picked at random, then find the probability of the two balls being 2 and 3 . (1) \(1 / 28\) (2) \(2 / 27\) (3) \(1 / 14\) (4) \(1 / 7\)

Short answer

Answer: The probability of drawing balls numbered 2 and 3 is \(\frac{1}{28}\).

Step-by-step solution

Find total number of combinations of drawing 2 balls

We can use the combination formula for this step: C(n, k) = n! / (k!(n-k)!) Where n is the total number of balls and k is the number of balls we want to pick. In this problem, n = 8 balls, and k = 2 balls. C(8, 2) = 8! / (2!(8-2)!) = 8! / (2!6!) = (8*7*6!)/(2!6!)= 28 So, there are 28 possible combinations of drawing 2 balls out of 8.

Identify the number of successful outcomes

In this case, our successful outcome is drawing balls 2 and 3. Since we're only picking 2 balls, there's only one successful outcome: drawing ball 2 and ball 3.

Calculate the probability

To find the probability of the successful outcome, we'll divide the number of successful outcomes by the total number of possible combinations: Probability =successful outcomes / total possible combinations = 1 / 28 Therefore, the probability of drawing balls 2 and 3 is: (1) \(\frac{1}{28}\)

Key concepts

Combination Formula

When solving problems that involve selecting items from a set without considering the order, the combination formula proves to be an indispensable tool. Let's understand the combination formula with an example. Imagine you have a collection of books, and you want to know in how many ways you can pick a certain number of books from the shelf. This is where the combination formula comes into play.

The combination formula is represented as follows: \begin{align*} C(n, k) = \frac{n!}{k!(n-k)!} \end{align*}In this formula, the terms have specific meanings:

• \( n \) represents the total number of items,
• \( k \) is the number of items to choose,
• Factorials (denoted by '!') signify the product of all positive integers up to that number.

For example, if you have 8 balls and want to know how many different pairs you can draw, you would plug the numbers into the combination formula like so: \begin{align*} C(8, 2) = \frac{8!}{2!(8-2)!} = 28. \end{align*}This calculation shows there are 28 unique ways to pick 2 balls out of 8 without considering the order. By mastering the combination formula, you unlock the power to compute the potential groupings in various scenarios, making it a fundamental concept in combinatorics.

Factorial Notation

The factorial notation is quite essential in combinatorics and probability calculations. It is represented by an exclamation mark (!) and is used to denote the product of an integer and all the integers below it down to one.Here's how it works:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• \( 3! = 3 \times 2 \times 1 = 6 \).

In the context of our previous example, \( 8! \) (read as 'eight factorial') would be the product of all positive integers from 1 to 8. Factorial notation is used in calculating combinations because it succinctly expresses the multiplications needed for these larger numbers.Understanding factorial notation is beneficial not only in computing probability but also in permutations, algebra and even in functions of calculus. It conveys a large amount of multiplication in a compact symbol, allowing for efficient calculation of complex problems.

Probability Calculation

Probability calculation involves determining the likelihood of an event occurring among a set of possible outcomes. It's expressed as a fraction or a percentage, representing the ratio of the favorable outcomes to the total number of possible outcomes.

In our example, to determine the probability of drawing two specific balls from a bag, we follow a simple process:

1. Identify the total number of possible outcomes, using the combination formula;
2. Determine the number of favorable outcomes;
3. Divide the number of favorable outcomes by the total outcomes to find the probability.

Thus, if you have one specific pair of balls in mind (like balls 2 and 3), and there is only one combination that matches this criterion, the probability will be \( \frac{1}{28} \). The practical applications of probability calculation extend beyond balls in a bag — it's used in fields as varied as finance, health, sports and even weather forecasting.

Outcomes in Probability

Understanding outcomes is essential when dealing with probability. An outcome is a possible result of an experiment or situation. When tossing a coin, there are two possible outcomes: heads or tails. In our exercise with the bag of balls, each pair of balls drawn represents a different outcome.

There are two types of outcomes in probability:

• Favorable outcomes: These are outcomes that fulfill the condition we are looking for.
• Possible outcomes: These include all the potential outcomes that can occur.

Being able to differentiate between these helps in calculating the probability of an event. The probability is directly proportional to the number of favorable outcomes and inversely proportional to the total possible outcomes. By identifying the correct type of outcome and using the other concepts we've discussed, like the combination formula and factorial notation, you can solve various probability problems with confidence.