91Ó°ÊÓ

For a biased die, the probabilities for different faces to turn up are \(\begin{array}{lcccccc}\text { Face } & : 1 & 2 & 3 & 4 & 5 & 6 \\ \text { Probability: } & 0.2 & 0.22 & 0.11 & 0.25 & 0.05 & 0.17\end{array}\) The die is tossed and you are told that either face 4 or face 5 has turned up. The probability that it is face 4 is (a) \(1 / 6\) (b) \(1 / 4\) (c) \(5 / 6\) (d) None

A coin is tossed twice. Find the probability distribution of the number of heads.

There are three urns \(A, B\) and \(C\). Urn \(A\) contains 4 white balls and 5 blue balls. \(\operatorname{Urn} B\) contains 4 white balls and 3 blue balls. \(\operatorname{Um} C\) contains 2 white balls and 4 blue balls. One ball is drawn from each of these urns. What is the probability that out of these three balls drawn, two are white balls and one is a blue ball. [CBSE-96]

Word UNIVERSITY is arranged randomly. Then the probability that both \(I\) does not come together is: [UPSEAT-2001] |(a) \(3 / 5\) (b) \(2 / 5\) (c) \(4 / 5\) (d) \(1 / 5\) Solution (c) Total number of ways \(=\frac{10 !}{2 !}\) Favourable number of ways for \(I\) come together is \(9 !\) Thus, probability that \(I\) come together $$ =\frac{9 ! \times 2 !}{10 !}=\frac{2}{10}=\frac{1}{5} $$

A bag contains 5 red and 7 black balls. Second bag contains 4 blue and 3 green balls. One ball is drawn from each bag. Find the probability for: \([\) MP-2005 (B)] (i) 1 red and 1 blue ball (ii) 1 green and 1 black ball

In a certain town, \(40 \%\) of the people have brown hair, \(25 \%\) have brown eyes and \(15 \%\) have both brown hair and brown eyes. If a person selected at random from the town has brown hair, the probability that he also has brown eyes is (a) \(1 / 5\) (b) \(3 / 8\) (c) \(1 / 3\) (d) \(2 / 3\)

A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, just two consecutive letters. TA are visible. The probability that the letters have come from CALCUTTA is (a) \(1 / 3\) (b) \(4 / 11\) (c) \(5 / 12\) (d) None of these (b) Let \(A\) : 'the event that letters came from TATANAGAR' B: 'the event that letters came from CALCUTTA' \(C\) : 'the event that two consecutive letters visible by \(T A^{\prime}\) Then \(P(A)=1 / 2, P(B)=1 / 2, P(C / B)=1 / 7\) Hence, by Bayes' theorem $$ P(B / C)=\frac{P(B) \times P(C / B)}{P(A) P(C / A)+P(B) P(C / B)}=\frac{4}{11} $$

A random variable \(X\) has the following probability distribution \(\begin{array}{rl}x: 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ P(X=x) & : a & 3 a & 5 a & 7 a & 9 a & 11 a & 13 a & 15 a & 17 a\end{array}\) Then value of \(a\) is (a) \(1 / 81\) (b) \(2 / 81\) (c) \(5 / 81\) (d) \(7 / 81\)

If the letters of the word ATTEMPT are written down at random. Find the probability if (i) all \(T\) s are together (ii) no two \(T\) s are together (a) (i) \(1 / 7\) (ii) \(2 / 7\) (b) (i) \(2 / 7\) (ii) \(1 / 7\) (c) (i) \(2 / 7\) (ii) \(3 / 7\) (d) (i) \(3 / 7\) (ii) \(1 / 7\)

Urn \(A\) contains 6 red and 4 black balls and urn \(B\) contains 4 red and 6 black balls. One ball is drawn at random from urn \(A\) and placed in urn \(B\). Then 1 ball is drawn at random from urn \(B\) and placed in urn \(A\). If 1 ball is now drawn at random from urn \(A\), the probability that it is found to be red is \([\) IIT-1988] (a) \(\frac{32}{55}\) (b) \(\frac{21}{55}\) (c) \(\frac{19}{55}\) (d) None of these (a) Let the events are \(R_{1}\) : 'a red ball is drawn from urn \(A\) and placed in \(B\) ' \(B_{1}:\) 'a black ball is drawn from urn \(A\) and placed in \(B\) ' \(R_{2}:\) 'a red ball is drawn from urn \(B\) and placed in \(A^{\prime}\) \(B_{2}:\) 'a black ball is drawn from urn \(B\) and placed in \(A\) 'Urn \(A\) contains 6 red and 4 black balls and urn \(B\) contains 4 red and 6 black balls. One ball is drawn at random from urn \(A\) and placed in urn \(B\). Then 1 ball is drawn at random from urn \(B\) and placed in urn \(A\). If 1 ball is now drawn at random from urn \(A\), the probability that it is found to be red is \([\) IIT-1988] (a) \(\frac{32}{55}\) (b) \(\frac{21}{55}\) (c) \(\frac{19}{55}\) (d) None of these (a) Let the events are \(R_{1}\) : 'a red ball is drawn from urn \(A\) and placed in \(B\) ' \(B_{1}:\) 'a black ball is drawn from urn \(A\) and placed in \(B\) ' \(R_{2}:\) 'a red ball is drawn from urn \(B\) and placed in \(A^{\prime}\) \(B_{2}:\) 'a black ball is drawn from urn \(B\) and placed in \(A\) '