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Probability

Short Answer Type

931. A man is known to speak the truth 5 out of 6 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

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Long Answer Type

932. Let there be three urns containing 1 white, 2 black, 3 red balls ; 2 white, 1 black, 1 red ball ; 4 white, 5 black, 3 red balls. One urn is chosen at random and two balls are drawn. These happen to be white and a red. What is the probability that they come from urn number 3.

Let E₁ , E₂, E₃ be events of selecting urns I, II, III respectively.
White Black Red
Urn I: 1 2 3
Urn II: 2 1 1
Urn III: 4 5 3

Let A be the event that the drawn balls are, one red and one white.
Since, the urns are equally likely
$straight P left parenthesis straight E subscript 1 right parenthesis space equals space 1 third comma space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 1 third comma space space straight P left parenthesis straight E subscript 3 right parenthesis space equals space 1 third$
∴ P( A/E₁) = Probability that one red and one white ball are drawn from first urn
$equals fraction numerator space straight C presuperscript 1 subscript 1 cross times straight C presuperscript 3 subscript 1 over denominator straight C presuperscript 6 subscript 2 end fraction space equals space 3 over 15 space equals space 1 fifth$
$straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis space equals space fraction numerator straight C presuperscript 1 subscript 1 cross times straight C presuperscript 2 subscript 1 over denominator straight C presuperscript 4 subscript 2 end fraction space equals 1 third comma space space space space straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis space equals space fraction numerator straight C presuperscript 3 subscript 1 cross times straight C presuperscript 4 subscript 1 over denominator straight C presuperscript 12 subscript 2 end fraction equals space 2 over 11$
$straight P left parenthesis straight E subscript 3 divided by straight A right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 3 right parenthesis. space space space straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis. space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis. space straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis space plus space straight P left parenthesis straight E subscript 3 right parenthesis. space straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis end fraction space space space space space space space space space space space space space space space space space equals space fraction numerator begin display style 1 third end style cross times begin display style 2 over 11 end style over denominator begin display style 1 third end style cross times begin display style 1 fifth end style plus begin display style 1 third end style cross times begin display style 1 third end style plus begin display style 1 third end style cross times begin display style 2 over 11 end style end fraction space equals fraction numerator begin display style 2 over 11 end style over denominator begin display style 1 fifth end style plus begin display style 1 third end style plus begin display style 2 over 11 end style end fraction space equals fraction numerator begin display style 2 over 11 end style over denominator begin display style fraction numerator 33 plus 55 plus 30 over denominator 165 end fraction end style end fraction$

$equals space 2 over 11 cross times 165 over 118 space equals 15 over 59$

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933. A pack of playing cards was found to contain only 51 cards. If the first 13 cards which are examined are all red, what is the probability that the missing card is black.

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934.

Coloured balls are distributed in four boxes as shown in the following table:

Box		Colour
	Black	White	Red	Blue
I	3	4	5	6
II	2	2	2	2
III	1	2	3	1
IV	4	3	1	5

A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III?

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935.

Suppose we have four boxes A, B, C and D containing coloured marbles as given below:

Box	Marble Colour
	Red	White	Black
A	1	6	3
B	6	2	2
C	8	1	1
D	0	6	4

One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red. what is the probability that it was drawn from box A? box B? box C?

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Multiple Choice Questions

936. Probability that A speaks truth is

4 over 5.

A coin is tossed. A reports that a head appears. The probability that actually there was head is

$4 over 5$
$1 half$
$1 fifth$
$1 fifth$

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Short Answer Type

937. A person plays a game of tossing a coin thrice. For each head, he is given Rs. 2 by the organiser of the game and for each tail, he has to give Rs. 1.50 to the organiser. Let X denote the amount gained or lost by the person. Show that X is a random variable and exhibit it as a function on the sample space of the experiment.

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938. An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X ? Is X a random variable?

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939. Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?

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940. A bag contains 2 white and 1 red balls. One ball is drawn at random and then put back in the box after noting its colour. The process is repeated again. If X denotes the number of red balls recorded in the two draws, describe X.

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