Three machines A, B, and C produce respectively 50%, 30%, and 20% of the total number of items of a factory. The percentage of defective output of these machines is 3%, 4%, and 5% respectively. If an item is selected at random, find the probability that the item is non-defective.

Answer:

the probability that a randomly selected item is non-defective is approximately 96.3%.

Step-by-step explanation:

This involves finding the probability of an item being non-defective for each machine and then combining these probabilities based on the machine's contribution to the total production.

Let's calculate it step by step:

Probability that an item from Machine A is non-defective:

The probability of a defective item from Machine A is 3%, so the probability of a non-defective item from Machine A is 100% - 3% = 97%.

Probability that an item from Machine B is non-defective:

The probability of a defective item from Machine B is 4%, so the probability of a non-defective item from Machine B is 100% - 4% = 96%.

Probability that an item from Machine C is non-defective:

The probability of a defective item from Machine C is 5%, so the probability of a non-defective item from Machine C is 100% - 5% = 95%.

Now, we need to consider the contribution of each machine to the total production:

Machine A produces 50% of the items.

Machine B produces 30% of the items.

Machine C produces 20% of the items.

To find the overall probability that a randomly selected item is non-defective, we'll use a weighted average:

Probability (Non-Defective) = (Probability from A * Fraction from A) + (Probability from B * Fraction from B) + (Probability from C * Fraction from C)

Probability (Non-Defective) = (97% * 50%) + (96% * 30%) + (95% * 20%)

Now, calculate the weighted average:

Probability (Non-Defective) = (0.97 * 0.50) + (0.96 * 0.30) + (0.95 * 0.20)

Probability (Non-Defective) = 0.485 + 0.288 + 0.19

Probability (Non-Defective) = 0.963

So, the probability that a randomly selected item is non-defective is approximately 96.3%.

The probability that an item randomly selected from the production of machines A, B, and C is non-defective is 0.963 or 96.3%.

The question is about calculating the probability of an item being non-defective in a factory production environment. Here is how you can find the solution:

1. Determine the probability that an item is produced by each machine and it isn't defective.
2. Machine A: Probability = 0.50 (proportion of total items) x 0.97 (proportion of non-defective items) = 0.485
3. Machine B: Probability= 0.30 x 0.96 = 0.288
4. Machine C: Probability = 0.20 x 0.95 = 0.19
5. Add the probabilities from each machine. This is valid because the events are mutually exclusive; an item can only be produced by one machine. Therefore, the total probability that an item randomly selected is non-defective is: 0.485 + 0.288 + 0.19 = 0.963 or 96.3%.

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