A problem is given to three students, whose chances of solving i

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Probability

Long Answer Type

871.

The probability of A solving a problem is $3 over 7$ and that of B solving it is $1 third.$ What is the probability that
(i) at least one of them will solve the problem?
(ii) Only one of them will solve the problem?

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Short Answer Type

872. A speaks truth in 60% cases and B in 90% cases. In what percentage of cases are they likely to contradict each other in stating the same fact.

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873. A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in narrating the same incident?

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874. The probability of hitting a target by three marksmen are

1 half comma space 1 third space and space 1 fourth

respectively. Find the probability that one and only one of them will hit the target when they fire simultaneously.

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875. An anti-aircraft gun can take a minimum of four shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

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876. A problem is given to three students, whose chances of solving it are $1 third comma space 1 fifth space and space 1 over 6$ respectively. What is the probability that exactly one of them may solve it?

Let A, B, C be the events that three students solve the problem.

therefore space space space space straight P left parenthesis straight A right parenthesis space equals space 1 third comma space space space straight P left parenthesis straight B right parenthesis space equals space 1 fifth comma space space straight P left parenthesis straight C right parenthesis space equals space 1 over 6

straight P left parenthesis straight A with bar on top right parenthesis space equals space 1 minus space straight P left parenthesis straight A right parenthesis space equals space 1 minus 1 third space equals space 2 over 3 comma space space straight P left parenthesis straight B with bar on top right parenthesis space equals space 1 minus 1 fifth space equals 4 over 5 comma space space straight P left parenthesis straight C with bar on top right parenthesis space equals space 1 minus 1 over 6 space equals 5 over 6

P(exactly one student solves the problem)

equals space straight P left parenthesis straight A space straight B with bar on top space straight C with bar on top space space space or space space space straight A with bar on top space straight B with bar on top space straight C with bar on top space or space straight A with bar on top space straight B with bar on top space straight C right parenthesis

equals space straight P left parenthesis straight A space straight B with bar on top space straight C with bar on top right parenthesis space plus space straight P left parenthesis straight A with bar on top space straight B space straight C with bar on top right parenthesis space plus space straight P left parenthesis straight A with bar on top space straight B with bar on top space straight C right parenthesis space equals space straight P left parenthesis straight A right parenthesis space straight P left parenthesis straight B with bar on top right parenthesis space straight P left parenthesis straight C with bar on top right parenthesis space plus space straight P open parentheses straight A with bar on top close parentheses space straight P left parenthesis straight B right parenthesis space straight P open parentheses top enclose straight C close parentheses space plus space straight P open parentheses straight A with bar on top close parentheses space straight P open parentheses straight B with bar on top close parentheses space straight P left parenthesis straight C right parenthesis equals space 1 third cross times 4 over 5 cross times 5 over 6 plus 2 over 3 cross times 1 fifth cross times 5 over 6 plus 2 over 3 cross times 4 over 5 cross times 1 over 6 equals space 20 over 90 plus 10 over 90 plus 8 over 90 space equals fraction numerator 20 plus 10 plus 8 over denominator 90 end fraction space equals 38 over 90 space equals 19 over 45

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877. A problem is given to three students, whose chances of solving it are

1 fourth comma space 1 third space and space 1 half

respectively. Find the probability that exactly one of them may solve it.

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878. The probabilities of A, B, C solving a problem are

1 third comma space 2 over 7 space and space 3 over 8

respectively. If all the three try to solve the problem simultaneously find the probability that exactly one of them will solve it.

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879. A problem in Mathematics is given to three students Dayanand, Ramesh and Naresh and whose cances of solving it are

1 half comma space 1 third comma space 1 fourth

respectively, what is the probability that the problem will be solved?

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Long Answer Type

880. The probability of A hitting a target is

4 over 5

and that of B hitting it is

2 over 3.

They both fire at the target. Find the probability that
(i) at least one of them will hit the target.
(ii) only one of them will hit the target.

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