Question

A manufacturer tests 1200 computers and finds that 9 of them have defects. Find the probability that a computer chosen at random has a defect. Predict the number of computers with defects in a shipment of 15,000 computers. Explain your reasoning.

Short answer

The probability that a computer selected at random has a defect is 0.0075 or 0.75%. Predicted defects in 15,000 computers are approximately 113.

Step-by-step solution

Calculate the Probability of One Defective Computer

In this case, probability (P) is calculated using the ratio of the number of defective computers (9) to the total number of computers tested (1200). This can be represented as: \[ P = \frac{9}{1200} \].

Simplify the Probability

After calculating the probability, it's important to simplify it to a lower form for easier understanding and further calculations. The probability P, when simplified, gives us: \[ P = 0.0075 \]

Predict Defective Computers in Shipment

The calculated probability can now be used to predict the number of defective computers in a shipment of 15,000 computers. This prediction can be made using the formula: \[ \text{Predicted Defects} = P * \text{Total Shipment} \]. The predicted number of defects is: \[ \text{Predicted Defects} = 0.0075 * 15000 \].

Simplify the Predicted Defects

After calculating, simplify the answer to get: \[ \text{Predicted Defects} = 112.5 \]. In real-world situations, we can't have a fraction of a computer, so we take the closest whole number, suggesting there may be around 113 defective computers in the shipment.

Key concepts

Defective Probability Calculation

Probability is a way to measure how likely an event is to happen. For defective probability calculation, it starts by looking at how many items are faulty out of the total tested. In this problem, 9 computers were found defective out of 1200 tested. To calculate the probability of finding a defective computer, we use the formula:

• Probability (\( P \)) = Number of defective items / Total number of items.

This gives us \( P = \frac{9}{1200} \), which tells us that for every computer tested, we have this fraction of a chance to find a defect. Calculating probability helps us understand the average likelihood of finding a defective computer.
By understanding this concept, we can evaluate risks and make informed decisions based on data.

Simplifying Probabilities

Once you calculate the initial probability, simplifying it makes it easier to interpret and use. In our example, the probability formula \( \frac{9}{1200} \) can be simplified to a decimal. To do this, divide the numbers:

• 9 ÷ 1200 = 0.0075

A simplified probability like 0.0075 is much easier to understand and compare with other probabilities.
Simplification not only helps in understanding but is also essential for subsequent calculations. For example, when predicting future occurrences, having a simple probability results in accurate and quick computations. Think of it as reducing fractions in math class—smaller numbers are less cumbersome to work with!

Prediction Using Probability

Probabilities are valuable for making predictions about larger samples or future events. With our calculated probability of 0.0075 for a defective computer, we can use this to predict defects in a shipment of any size.
To predict defects in a shipment of 15,000 computers:

• Use the formula: Predicted Defects = Probability × Total number in shipment.

This is calculated as:

• Predicted Defects = 0.0075 × 15000.

The result, 112.5, suggests that in a large shipment, you might expect about 113 defective computers. Predictions like these are useful in planning and resource allocation.
Knowing the potential number of defects helps companies prepare for replacements and manage customer expectations effectively.

Real-world Application of Probability

Probability isn't just a concept for textbooks; it's widely used in various real-world applications. Knowing how likely defects are in products allows manufacturers to control quality and ensure high standards. This application of probability extends beyond just computers to any manufacturing context where quality control is crucial.
Using probabilities enables businesses to:

• Plan for adequate inventory and replacement parts.
• Manage risks by predicting issues before they arise.
• Improve customer satisfaction by minimizing defective products.

Beyond manufacturing, probabilities also play a role in everyday decisions such as predicting weather, managing finances, and evaluating risks in various fields. Understanding these concepts empowers individuals and businesses to make better, informed decisions.