Problem 62 - Fundamentals of Mathematics

Question

State the numerator and denominator and write in words each of the fractions appearing in the statements for the following 10 problems. In a box that contains eight computer chips, five are known to be good and three are known to be defective. If three chips are selected at random, the probability that all three are defective is \(\frac{1}{56}\).

Step-by-Step Solution

Answer

The probability that all three chips are defective is 'one in fifty-six'.

Step 1: Identify the total number of chips (denominator)

The box contains a total of 8 computer chips, which forms the denominator of the fraction when calculating the probability of any event regarding these chips.

Step 2: Identify the goal event (numerator)

The specific event of interest is selecting 3 defective chips out of the 8. We need to find how many ways this particular combination can occur.

Step 3: Calculate numerator using combinations

Calculate number of ways to choose 3 defective chips from the 3 available. This is represented by the combination formula \( C(3,3) \), which equals 1.

Step 4: Construct the fractionated form

To find the probability that all 3 selected chips are defective, divide the number of favorable outcomes (1) by the total possible outcomes.

Step 5: Interpret the probability fraction in words

The probability fraction \( \frac{1}{56} \) in words is: 'one in fifty-six'. It signifies that 1 out of every 56 random selections of 3 chips will have all defective chips.

Key Concepts

Denominator

In fractions, the denominator is a crucial element. It's the number on the bottom of a fraction, and it represents the total number of equal parts into which something is divided. In our exercise, the denominator is connected to the total number of computer chips. Since there are 8 chips in the box, our denominator is 8 when calculating probabilities related to these chips.
The role of the denominator helps to understand how likely an event is concerning the whole set. It divides the context or the "whole" part of the situation.

• If you have a full pizza cut into 8 slices, each slice would be \(\frac{1}{8}\) of the whole pizza;
• Similarly, the 8 in our exercise means we're considering all 8 chips together.

Understanding the denominator aids in seeing the big picture and setting the stage for calculating probabilities.

Numerator

The numerator is the top number in a fraction, and it shows how many parts of the whole are being considered. In the context of our exercise, the numerator represents the number of favorable outcomes. It focuses on the specific event we are analyzing — selecting 3 defective chips.
The numerator stems from the selection or event part you're interested in, forming the fraction's "focus."

• For example, if you eat 3 slices of an 8-slice pizza, 3 becomes your numerator, meaning you've eaten \(\frac{3}{8}\) of the pizza.
• In the chip problem, determining how many ways you could draw 3 defective chips from those available gives the numerator. Here, only 1 way exists, so our numerator is 1.

Thus, the numerator narrows down the probability fraction to spotlight precisely what you want to find out.

Probability

Probability represents the likelihood of an event happening. Fractionally, it is the ratio of favorable outcomes (numerator) to the total possible outcomes (denominator). In this exercise, we want the probability that all selected chips are defective.
To find probability:

• Identify how many ways the event (all chips defective) can occur. This is your numerator.
• Identify total outcomes possible. This forms your denominator.

The calculation: Probability \(= \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}}\).
The given probability \(\frac{1}{56}\) means that, out of 56 attempts or tries, you'd expect this specific outcome (all chips defective) to happen once. This concept supports statistical predictions and decision-making based on likely outcomes.

Combinations

Combinations are a mathematical way to determine how many different groups or sets you can form from a larger set without regard to the order of selection. They're crucial in probability because they help establish the numerator in our probability fraction.
In our exercise, we use combinations to determine how many ways we can choose all defective chips from the set available.

• The formula used for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!}\), where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose.
• In our case, choosing 3 defective chips from 3 gives us \( C(3,3) = 1 \), which is straightforward since it's the whole set.

Combinations illustrate potential groupings within a context, fundamental for calculating probabilities without considering different orderings in chosen items.