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The probability that there is at least one error in an account statement prepared by 3 persons A, B and C are 0.2, 0.3 and 0.1 respectively. If A, B and C prepare 60, 70 and 90 such statements, then the expected number of correct statements
- a)170
- b)176
- c)178
- d)180
Correct answer is option 'C'. Can you explain this answer?
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The probability that there is at least one error in an account stateme...
Given:
The probability of an error in statement prepared by A, B and C are 0.2, 0.3 and 0.1 respectively.
A prepares 60 statements, B prepares 70 statements and C prepares 90 statements.
To find:
The expected number of correct statements.
Solution:
Let X be the number of correct statements.
Then, the number of incorrect statements will be (60 - X) for A, (70 - X) for B and (90 - X) for C.
Therefore, the probability of all statements being incorrect will be:
P(X = 0) = P(A is incorrect) × P(B is incorrect) × P(C is incorrect)
= 0.2 × 0.3 × 0.1
= 0.006
The probability of exactly one statement being correct will be:
P(X = 1) = P(A is correct) × P(B is incorrect) × P(C is incorrect) +
P(A is incorrect) × P(B is correct) × P(C is incorrect) +
P(A is incorrect) × P(B is incorrect) × P(C is correct)
= (0.8) × (0.3) × (0.1) + (0.2) × (0.7) × (0.1) + (0.2) × (0.3) × (0.9)
= 0.214
Similarly, we can find the probabilities of X = 2 and X = 3 as:
P(X = 2) = P(A is correct) × P(B is correct) × P(C is incorrect) +
P(A is correct) × P(B is incorrect) × P(C is correct) +
P(A is incorrect) × P(B is correct) × P(C is correct)
= (0.8) × (0.7) × (0.1) + (0.8) × (0.3) × (0.9) + (0.2) × (0.7) × (0.9)
= 0.435
P(X = 3) = P(A is correct) × P(B is correct) × P(C is correct)
= (0.8) × (0.7) × (0.9)
= 0.504
The expected number of correct statements can be calculated as:
E(X) = Σ(xi × pi)
= (0 × 0.006) + (1 × 0.214) + (2 × 0.435) + (3 × 0.504)
= 1.78
Therefore, the expected number of correct statements is 178. Hence, option (c) is the correct answer.
The probability of an error in statement prepared by A, B and C are 0.2, 0.3 and 0.1 respectively.
A prepares 60 statements, B prepares 70 statements and C prepares 90 statements.
To find:
The expected number of correct statements.
Solution:
Let X be the number of correct statements.
Then, the number of incorrect statements will be (60 - X) for A, (70 - X) for B and (90 - X) for C.
Therefore, the probability of all statements being incorrect will be:
P(X = 0) = P(A is incorrect) × P(B is incorrect) × P(C is incorrect)
= 0.2 × 0.3 × 0.1
= 0.006
The probability of exactly one statement being correct will be:
P(X = 1) = P(A is correct) × P(B is incorrect) × P(C is incorrect) +
P(A is incorrect) × P(B is correct) × P(C is incorrect) +
P(A is incorrect) × P(B is incorrect) × P(C is correct)
= (0.8) × (0.3) × (0.1) + (0.2) × (0.7) × (0.1) + (0.2) × (0.3) × (0.9)
= 0.214
Similarly, we can find the probabilities of X = 2 and X = 3 as:
P(X = 2) = P(A is correct) × P(B is correct) × P(C is incorrect) +
P(A is correct) × P(B is incorrect) × P(C is correct) +
P(A is incorrect) × P(B is correct) × P(C is correct)
= (0.8) × (0.7) × (0.1) + (0.8) × (0.3) × (0.9) + (0.2) × (0.7) × (0.9)
= 0.435
P(X = 3) = P(A is correct) × P(B is correct) × P(C is correct)
= (0.8) × (0.7) × (0.9)
= 0.504
The expected number of correct statements can be calculated as:
E(X) = Σ(xi × pi)
= (0 × 0.006) + (1 × 0.214) + (2 × 0.435) + (3 × 0.504)
= 1.78
Therefore, the expected number of correct statements is 178. Hence, option (c) is the correct answer.
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The probability that there is at least one error in an account statement prepared by 3 persons A, B and C are 0.2, 0.3 and 0.1 respectively. If A, B and C prepare 60, 70 and 90 such statements, then the expected number of correct statementsa)170b)176c)178d)180Correct answer is option 'C'. Can you explain this answer?
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