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Question
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is ______.
Options
`1/4`
`1/3`
`1/2`
`3/4`
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Solution 1
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is `underlinebb(3/4)`.
Explanation:
Let A, B, C be the respective events of solving the problem.
Then, P(A) = `1/2`, P(B) = `1/3` and P(C) = `1/4`.
Here, A, B, C are independent events.
Problem is solved if at least one of them solves the problem.
Required probability is
= P(A ∪ B ∪ C)
= `1 - P(overlineA)P(overlineB)P(overlineC)`
= `1 - (1 - 1/2)(1 - 1/3)(1 - 1/4)`
= `1 - 1/4`
= `3/4`.
Solution 2
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is `underlinebb(3/4)`.
Explanation:
The problem will be solved if one or more of them can solve the problem.
The probability is
`P(Aoverline(BC)) + P(overlineABoverlineC) + P(overline(AB)C) + P(ABoverlineC) + P(AoverlineBC) + P(overlineABC) + P(ABC)`
= `1/2. 2/3. 3/4 + 1/2. 1/3. 3/4 + 1/2 . 2/3. 1/4 + 1/2. 1/3. 3/4 + 1/2. 2/3. 1/4 + 1/2. 1/3. 1/4 + 1/2. 1/3. 1/4`
= `3/4`.
Solution 3
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is `underlinebb(3/4)`.
Explanation:
Let us think quantitively.
Let us assume that there are 100 questions given to A.
A solves `1/2 xx 100` = 50 questions then remaining 50 questions is given to B and B solves `50 xx 1/3` = 16.67 questions.
Remaining `50 xx 2/3` questions is given to C and C solves `50 xx 2/3 xx 1/4` = 8.33 questions.
Therefore, number of questions solved is 50 + 16.67 + 8.33 = 75.
So, required probability is `75/100 = 3/4`.
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