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(1)
Example 1 : There are 11 tickets in a box bearing numbers 1 to 11.
Three tickets are drawn one after the other without replacement.
Find the probability that they are drawn in the order bearing (i) even,
odd, even number, (ii) odd, odd, even number.
(M.U. 1997)
Sol. : This is an example on conditional probability. Out of 11 tickets, 5
are even and 6 are odd.
=
(2)
Example 2 : Four roads lead away from a jail. A prisoner trying to
escape from the jail selects a road at random. If road A is selected, the
probability of escaping is 1/8, for road B, it is 1/6, for road C it is 1/4
and and for road D it is 9/10. What is the probability that a prisoner
will succeed in escaping from the jail?
(M.U. 2005)
Sol. : Let E= Success in escaping.
1/4,
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1/4,
1/8,
1/6,
1/4,
9/10,
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(3)
Example 3 : Ina certain test there are multiple choice questions.
There are four possible answers to each question and one of them is
correct. An intelligent student can solve 90% questions correctly by
reasoning and for the remaining 10% questions he gives answers by
guessing. A weak student can solve 20% questions correctly by
reasoning and for the remaining 80 % questions he gives answers by
guessing. An intelligent students get the correct answer, what is the
probability that he was guessing ?
(M.U. 2004)
Sol. : Consider the intelligent student.
Let
Required Probability
(4)
Example 4 : A certain test for a particular cancer is known to be
95% accurate. A person submits to the test and the result is positive.
Suppose that a person comes from a population of 100,000 where
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2000 people suffer from that disease. What can we conclude about
the probability that the person under test has particular cancer ?
(M.U. 2006)
Sol. : we have
Probability a person has the cancer
Probability that a person does not have the cancer
Test is positive when a person has cancer
Test is positive when person does not have a cancer
(5)
Example 5 : A bag contains 7 red and 3 black balls and another bag
contains 4 red and 5 black balls. One ball is transferred from the first
bag to the second bag and then a ball is drawn from the second bag. If
this ball happens to be red, find the probability that black ball was
transferred.
(M.U. 2002)
Sol. : We have
Probability of transferring black ball
Probability of now drawing a red ball
Probability of transferring red ball
Probability of now drawing red ball
Required Probability
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(6)
Example 6 : Write down the probability distribution of the sum of
numbers appearing on the toss of two unbiased dice. (M.U. 2004,06)
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
Sol. : When two dice are thrown, we get the sum of numbers as
shown on the above by slant dotted lines.
It is easy to see that the sum 2 appears once, 3 twice, 4 thrice
etc. the probability of each single event is 1/36.
The probability distribution obtained is shown below.
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Example 7 : The probability mass function of a random variable X is
zero accept at the points X=0, 1, 2. At this points it has the values
Determine c, (ii) Find
(M.U. 2001)
Sol. : Since
(The other values are not admissible)
The probability distribution is
(7)
Example 8 : Let x be a continuous random variable with
probability distribution
Elevation k and find
Sol. : Since the total probability is one
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(M.U. 1998,2004)
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Example 9 : Let X be a continuous random variable with p.d.f.
Find k and determine a number b such
that
(M.U.2003, 11)
Sol. : Since
(8)
Since, the total probability is 1 and
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(9)
Example 10 : The probability that a person will die in the time
interval
is given by
Where,
Find (i) the probability that Mr. X will die between the ages 60
and 70. (ii) the probability that he will die between the ages 60 and 70, given
that he has survived upto age 60.
(M.U. 2005)
Sol. : (i)
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(i)
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