A bolt manufacturing factory has three products A, B and C. 50% and 30% of the products are A and B type respectively and remaining are C type. Then probability that the product A is defective is 4%, that of B is 3% and that of C is 2%. A product is picked randomly picked and found to be defective, then the probability that it is type C.
A bolt manufacturing factory has three products A, B and C. 50% and 30% of the products are A and B type respectively and remaining are C type. Then probability that the product A is defective is 4%, that of B is 3% and that of C is 2%. A product is picked randomly picked and found to be defective, then the probability that it is type C.
Show Hint
Bayes' Theorem is a powerful tool for calculating conditional probabilities. In problems like this, where different types of events (products in this case) have different probabilities of defects, Bayes' Theorem allows us to update our probability after observing a new event (a defective product). Always ensure that you calculate the total probability of the event before applying the formula.
- \(\frac{4}{33}\)
- \(\frac{1}{33}\)
- \(\frac{2}{33}\)
- \(\frac{9}{33}\)
The Correct Option is A
Solution and Explanation
Step 1: Define the Given Probabilities
We are given the following probabilities:
- Probability that the product is type A: \( P(A) = 0.50 \)
- Probability that the product is type B: \( P(B) = 0.30 \)
- Probability that the product is type C: \( P(C) = 1 - P(A) - P(B) = 0.20 \)
- Probability that A is defective: \( P(D|A) = 0.04 \)
- Probability that B is defective: \( P(D|B) = 0.03 \)
- Probability that C is defective: \( P(D|C) = 0.02 \)
Step 2: Use Bayes' Theorem
We need to find the probability that the product is type C given that it is defective. This can be calculated using Bayes' Theorem:
Step 3: Calculate Total Probability of Defect
The total probability of picking a defective product is given by:
Substituting the given values:
Step 4: Calculate \( P(C|D) \)
Now, substitute the values into Bayes' Theorem:
Final Answer:
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Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.