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Probability

Long Answer Type

821.
Prove that if E and F are independent events, then so are the events E and F'.

Since E and F are independent.
∴ P(E ∩ F) = P(E). P(F)
From the Venn diagram in the figure, it is clear that E ∩ F and E ∩ F' are mutually exclusive events and also
$straight E space equals space left parenthesis straight E space intersection space straight F right parenthesis thin space left parenthesis straight E space intersection space straight F apostrophe right parenthesis.$
$therefore space space space space straight P left parenthesis straight E right parenthesis space equals space straight P left parenthesis straight E intersection straight F right parenthesis space plus space straight P left parenthesis straight E intersection straight F apostrophe right parenthesis rightwards double arrow space space straight P left parenthesis straight E space intersection space straight F apostrophe right parenthesis space equals space straight P left parenthesis straight E right parenthesis space minus space straight P left parenthesis straight E space intersection space straight F right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight P left parenthesis straight E right parenthesis space minus space straight P left parenthesis straight E right parenthesis. space straight P left parenthesis straight F right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets space space space space space space space space space space space space space space space space space space space space space space space space equals space straight P left parenthesis straight E right parenthesis space left square bracket 1 minus space straight P left parenthesis straight F right parenthesis right square bracket space space space space space space space space space space space space space space space space space space space space space space space space equals space straight P left parenthesis straight E right parenthesis. space straight P left parenthesis straight F apostrophe right parenthesis$
∴ E and F' are independent.

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Short Answer Type

822. If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by 1 - P(A') P(B').

93 Views

Multiple Choice Questions

823.

Two events A and B will be independent, if

A and B are mutually exclusive
P(A'B') = [1 - P(A)] [1 - P(B)]
P(A) = P(B)
P(A) = P(B)

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Short Answer Type

824. An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

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825. Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

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826. An unbiased die is thrown. Find the probability of getting 4, 5 or 6 in the first throw and 1, 2, 3 or 4 in the second throw.

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827. Two dice are thrown. Find the probability of getting an odd number on the first die and a multiple of 3 on the other.

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828. A die is tossed twice. Find, the probability of ‘a number greater than 4’ on each dice.

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829. A coin is tossed. If head comes up, a die is thrown but if tail comes up, the coin is tossed again. Find the probability of obtaining two tails.

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830. A coin is tossed. If head comes up, a die is thrown but if tail comes up, the coin is tossed again. Find the probability of obtaining head and a number 6.

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