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.
Similar Questions
A coin is biased so that heads comes up thrice as likely as tails. In four independent tosses of the coin, what is probability of getting exactly three heads ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
In a class, there are n students including the students P and Q. What is the probability that P and Q sit together if seats are assigned randomly?
What is the probability that Q and U sit together?
What is the probability that boys and girls sit alternatively?
What is the probability that P and Q take the two end positions?
A bag contains 20 books out of which 5 are defective. If 3 of the books are selected at random and removed from the bag in succession without replacement, then what is the probability that all three books are defective?
If the probability of simultaneous occurrence of two events A and B is p and the probability that exactly one of A, B occurs is q, then which of the following is/are correct/
1) P(A̅) + P(B̅) = 2 – 2p – q
2) P(A̅ ∩ B̅) = 1 – p – q
Select the correct answer using the code given below:
Important Questions from Probability
Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is 12?
If a box contains 3 white cushions, 4 red cushions and 5 blue cushions, what is the probability of selecting a white or blue cushion?
A. 2/3
B. 3/4
C. 1/4
D. 1/9Statements followed by some conclusions are given below.
Statements:
1. A bag has 2 white, 3 black, 4 red and 6 green balls.
2. 1 ball selected at random from the bag.
Conclusions:
I. The probability that a black ball is selected is 1/5
II. The probability that a red ball is selected is 6/15
Find which of the conclusions logically follows from the given statement
A. Only conclusion I follows.
B. Only conclusion II follows.
C. Both I and II follow.
D. Neither I nor II follows.
In a shooting test, the probabilities of hitting the target are 1/2 for A, 2/3 for B and 3/4 for C. If they fire at the same target, what is the probability that only one of them hits the target?
A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:
