A machine has three parts, A, B and C, whose chances of being defective are 0.02, 0.10 and 0.05 respectively. The machine stops working if any one of the parts becomes defective. What is the probability that the machine will not stop working?
0.84
Understanding Machine Reliability Probability
The problem asks for the probability that a machine will continue working. The machine has three independent parts, A, B, and C. The machine stops working if any one of these parts becomes defective. Therefore, for the machine to continue working (not stop working), all three parts must be non-defective.
Given Probabilities of Defective Parts
We are given the probabilities that each individual part is defective:
- Probability that Part A is defective, \(P(A_{defective}) = 0.02\).
- Probability that Part B is defective, \(P(B_{defective}) = 0.10\).
- Probability that Part C is defective, \(P(C_{defective}) = 0.05\).
Calculating Probability of Parts Not Being Defective
If the probability of a part being defective is known, the probability of that part *not* being defective (i.e., working correctly) is 1 minus the probability of it being defective. This is because a part is either defective or not defective, and these are mutually exclusive and exhaustive events.
- Probability that Part A is NOT defective, \(P(A_{not\ defective}) = 1 - P(A_{defective}) = 1 - 0.02 = 0.98\).
- Probability that Part B is NOT defective, \(P(B_{not\ defective}) = 1 - P(B_{defective}) = 1 - 0.10 = 0.90\).
- Probability that Part C is NOT defective, \(P(C_{not\ defective}) = 1 - P(C_{defective}) = 1 - 0.05 = 0.95\).
We can summarize these probabilities in a table:
| Part | Probability of being Defective | Probability of NOT being Defective |
|---|---|---|
| A | 0.02 | 0.98 |
| B | 0.10 | 0.90 |
| C | 0.05 | 0.95 |
Calculating Probability Machine Will Not Stop Working
The machine stops working if *any* part is defective. This means the machine will *not* stop working only if *all* parts are *not* defective. Since the parts are independent, the probability of all three parts being non-defective is the product of the individual probabilities that each part is non-defective.
Let \(E\) be the event that the machine will not stop working.
\(P(E)\) = \(P(\text{A not defective AND B not defective AND C not defective})\)
Since the events are independent:
\(P(E) = P(A_{not\ defective}) \times P(B_{not\ defective}) \times P(C_{not\ defective})\)
Substitute the probabilities we calculated:
\(P(E) = 0.98 \times 0.90 \times 0.95\)
First, calculate \(0.98 \times 0.90\):
\(0.98 \times 0.90 = 0.882\)
Now, multiply the result by \(0.95\):
\(0.882 \times 0.95 = 0.8379\)
The probability that the machine will not stop working is 0.8379.
Comparing this value with the given options, 0.8379 is closest to 0.84.
Final Answer Derivation
The probability that the machine will not stop working is 0.8379, which rounds to 0.84.
Revision Table: Probability of Machine Parts
| Component | Failure Probability | Working Probability |
|---|---|---|
| Part A | 0.02 | \(1 - 0.02 = 0.98\) |
| Part B | 0.10 | \(1 - 0.10 = 0.90\) |
| Part C | 0.05 | \(1 - 0.05 = 0.95\) |
| Machine Working (All parts working) | N/A | \(0.98 \times 0.90 \times 0.95 = 0.8379\) |
Additional Information on Independent Events
In probability theory, two events are considered independent if the occurrence of one event does not affect the probability of the other event occurring. If events E1, E2, ..., En are independent, the probability that all of them occur is the product of their individual probabilities:
\(P(E1 \text{ and } E2 \text{ and } ... \text{ and } En) = P(E1) \times P(E2) \times ... \times P(En)\)
In this machine problem, the question states that the chances of each part being defective are given, implying these are inherent probabilities for each part and the failure of one part does not influence the probability of another part failing. Therefore, we treat the events "Part A is not defective," "Part B is not defective," and "Part C is not defective" as independent events.
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Important Questions from Probability
- If one event is unaffected by the outcome of another event, the two events are said to be:
1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive - Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \dots, 2026\}$.
Which of the following is the probability that the product of all the elements of $M$ is even? - A fair coin is tossed 6 times. The probability of getting at least 5 heads is _________________. (rounded off to two decimal places)
- Consider a random experiment of rolling a fair die. The probability of getting an odd number or a number less than 4 is ______ (Rounded off to two decimal places).
- In a null hypothesis significance test, the significance level $\alpha$ was set to 0.01. For which one or more of the following $p$ values would the null hypothesis be rejected?
