A manufacturer makes two models of an item: model I, which accounts for 80% of unit sales, and model II, which accounts for 20% of unit sales. Because of defects, the manufacturer has to replace (or exchange) 10% of its model I and 18% of its model II. If a model is selected at random, find the probability that it will be defective. The answer is .116, I just can't figure out how to work out the problem. Can anyone please help?
.80*.10 + .20*.18 = .116
Sure, I'd be happy to help! But before we dive into the calculations, let's start with a little humor to lighten the mood:
Why was the math book sad?
Because it had too many problems!
Now, onto the problem. To find the probability that a randomly selected model will be defective, we need to consider both the individual probabilities of each model being defective and their respective probabilities of being selected.
Let's break it down step by step:
Step 1: Calculate the probability of selecting model I and model II.
- Given that model I accounts for 80% (or 0.8) of unit sales and model II accounts for 20% (or 0.2), we have the following probabilities:
P(model I) = 0.8
P(model II) = 0.2
Step 2: Calculate the probability of each model being defective.
- We know that 10% (or 0.10) of model I and 18% (or 0.18) of model II are defective. Therefore, the probabilities are:
P(defective | model I) = 0.10
P(defective | model II) = 0.18
Step 3: Use the probabilities from steps 1 and 2 to calculate the overall probability of selecting a defective model.
- We can apply the law of total probability here:
P(defective) = P(defective | model I) * P(model I) + P(defective | model II) * P(model II)
Plugging in the values, we get:
P(defective) = 0.10 * 0.8 + 0.18 * 0.2
= 0.08 + 0.036
= 0.116
And there you have it! The probability of selecting a defective model is 0.116, or 11.6%.
I hope this helps! If you have any more questions or need further assistance, feel free to ask.
To find the probability that a randomly selected model will be defective, we can use the law of total probability.
Let's define:
A = event that model I is selected
B = event that model II is selected
D = event that the selected model is defective
We are given the following information:
P(A) = 80% = 0.8 (probability of selecting model I)
P(B) = 20% = 0.2 (probability of selecting model II)
P(D|A) = 10% = 0.1 (probability of defect given model I is selected)
P(D|B) = 18% = 0.18 (probability of defect given model II is selected)
The probability of selecting a defective model can be calculated as:
P(D) = P(A) * P(D|A) + P(B) * P(D|B)
= 0.8 * 0.1 + 0.2 * 0.18
= 0.08 + 0.036
= 0.116
Therefore, the probability that a randomly selected model will be defective is 0.116 (or 11.6%).
To find the probability that a randomly selected model will be defective, we need to consider two things:
1. The probability of selecting each model.
2. The probability of a selected model being defective.
Let's break down the problem step-by-step:
Step 1: Calculate the probability of selecting model I and model II.
- Given that model I accounts for 80% of unit sales, its probability of selection would be 0.80.
- Similarly, model II, accounting for 20% of unit sales, would have a probability of 0.20.
Step 2: Calculate the probability of selecting a defective model within each category.
- For model I, 10% of the units are defective. This means the probability of selecting a defective model I is 0.10 * 0.80 (probability of selecting model I).
- For model II, 18% of the units are defective. The probability of selecting a defective model II is 0.18 * 0.20 (probability of selecting model II).
Step 3: Calculate the overall probability of selecting a defective model.
- To do this, we need to add up the probabilities of selecting a defective model from each category.
- So, the overall probability is: (0.10 * 0.80) + (0.18 * 0.20) = 0.08 + 0.036 = 0.116.
Therefore, the probability of selecting a defective model is 0.116 or 11.6%.
Feel free to ask for further clarification if needed!