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Q. 19.     If A and B are any two events such that $\inline P(A)+P(B)-P(A \; and\; B)=P(A),$ then

(A) $\inline P(B\mid A)=1$

(B) $\inline P(A\mid B)=1$

(C) $\inline P(B\mid A)=0$

(D) $\inline P(A\mid B)=0$

Option B is correct.

Q. 18.     If $\inline P(A\mid B)> P(A)$,  then which of the following is correct :

(A) $\inline P(B\mid A)< P(B)$

(B) $\inline P(A\cap B)< P(A).P(B)$

(C) $\inline P(B\mid A)> P(B)$

(D)$\inline P(B\mid A)= P(B)$

Option C is  correct.

Q. 17.     If A and B are two events such that $\inline P(A\neq 0)$ and$\inline P(B\mid A)=1,$ then

Choose the correct answer of the following:

(A)  $\inline A\subset B$

(B)  $\inline B\subset A$

(C)  $\inline B=\phi$

(D)  $\inline A=\phi$

A and B are two events such that  and                                                                                                  Option A is correct.

Q. 16  Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

Let E1 and E2 respectively denote the event that red ball is transfered from bag 1 to bag 2 and a black ball is transfered from bag 1 to bag2.             and       Let A be the event that ball drawn is red. When a red ball is transfered from bag 1 to bag 2. When a black ball is transfered from bag 1 to bag 2.

Q. 15  An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:

P(A fails) = $0.2$

P(B fails alone) = $0.15$

P(A and B fail) = $0.15$

Evaluate the following probabilities

(ii) $\inline P(A\; fails \; alone)$

Let event in which A fails and B fails be

Q. 15.     An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:

P(A fails) =$\inline 0.2$

P(B fails alone) = $\inline 0.15$

P(A and B fail) = $\inline 0.15$

Evaluate the following probabilities

(i) $\inline P(A \; fails\mid B\; has\; failed)$

Let event in which A fails and B fails be

Q. 14.     If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$ ).

Total number of determinant of second order with each element being 0 or 1 is  The values of determinant is positive in the following cases          Probability is

Q. 13  Assume that the chances of a patient having a heart attack is $\inline 40^{o}/_{o}.$ It is also assumed that a meditation and yoga course reduce the risk of heart attack by $\inline 30^{o}/_{o}$ and prescription of certain drug reduces its chances by $\inline 25^{o}/_{o}.$At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that  the patient followed a course of meditation and yoga?

Let A,E1, E2 respectively denote the event that a person has a heart break, selected person followed the course of yoga and meditation , and the person adopted      the drug prescription.      the probability that the patient followed a course of meditation and yoga is

Q. 12 Suppose we have four boxes A,B,C and D containing coloured marbles as given below:

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 C?

Let R be event of drawing red marble. Let   respectivly denote event of selecting box A,B,C. Total marbles = 40 Red marbles =15      Probability of drawing red marble from box C  is

Q. 12.     Suppose we have four boxes A,B,C and D containing coloured marbles as given below:

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 B?

Let R be event of drawing red marble. Let   respectivly denote event of selecting box A,B,C. Total marbles = 40 Red marbles =15      Probability of drawing red marble from box B  is

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

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 ?

' Let R be the event of drawing red marble. Let   respectively denote the event of selecting box A, B, C. Total marbles = 40 Red marbles =15      Probability of drawing red marble from box A  is

Q. 11  In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins / loses.

In a throw of die, probability of getting six  = P                     probability of not getting six  = q               There are three cases : 1. Gets six in the first throw, required probability is       The amount he will receive is Re. 1 2.. Does not gets six in the first throw and gets six in the second throw, then the probability                                                           ...

Q. 10 How many times must a man toss a fair coin so that the probability of having at least one head is more than $90^{o}/_{o}$ ?

Let the man toss coin n times. Probability of getting head in first toss = P                                                                                                                                                                                                                                                                                                                      The...

Q. 9  An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes.

Probability of success is twice the probability of failure. Let probability of failure be X then Probability of success = 2X     Sum of probabilities is 1.                             Let                    and      Let X be random variable that represent the number of success in six trials.                                                                                                      ...

Q. 8  If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?

In a leap year, there are 366 days. In 52 weeks, there are 52 Tuesdays. The probability that a leap year will have 53 Tuesday is equal to the probability that the remaining 2 days are Tuesday. The remaining 2 days can be : 1. Monday and Tuesday 2. Tuesday and Wednesday 3. Wednesday and Thursday 4. Thursday and Friday 5.friday and Saturday 6.saturday and Sunday 7.sunday and Monday  Total cases =...

Q.7 A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Probability of 6 in a throw of die =P                                                                          Probability that 2 sixes come in first five throw of die :                                                                                    Probability that third six comes in sixth throw :                                                                                              ...

Q. 6  In a hurdle race, a player has to cross $10$  hurdles. The probability that he will clear each hurdle is $\frac{5}{6}$ . What is the probability that he will knock down fewer than $2$ hurdles?

Let p and q respectively be probability that the player will clear and knock down the hurdle.      Let X represent random variable that represent number of times the player will knock down the hurdle.

Q. 5   An urn contains 25 balls of which 10 balls bear a mark $\inline 'X'$ and the remaining 15 bear a mark $\inline 'Y'.$ A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that

(iv) the number of balls with $\inline 'X'$ mark and $\inline 'Y'$ mark will be equal.

Q. 5  An urn contains 25 balls of which 10 balls bear a mark  $'X'$  and the remaining 15 bear a mark $'Y'.$ A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that

(iii) at least one ball will bear $'Y'$ mark.

Q. 5  An urn contains $25$ balls of which $10$ balls bear a mark $'X'$  and the remaining $15$ bear a mark  $'Y'.$  A ball is drawn at random from the urn, its mark is noted down and it is replaced. If $6$  balls are drawn in this way, find the probability that

(ii) not more than $2$ will bear $'Y'$  mark.

Total balls in urn = 25 Balls bearing mark 'X' =10 Balls bearing mark 'Y' =15   balls are drawn with replacementt. Let Z be random variable that represents number of balls with Y mark on them in trial. Z has binomail distribution with n=6.
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