Total Probability Theorem

Q. Out of all the arrangements that can be made taking 5 letters at a time of the word BRILLIANT one is chosen at random. The probability that this will have 5 distinct letters is

1. $\frac{255}{702}$
2. $\frac{257}{502}$
3. $\frac{252}{507}$
4. $\frac{522}{705}$

Q. There are two bags, one of which contains 5 black and 4 white balls while the other contains 3 black and 6 white balls. A die is thrown. If it shows up 1 or 3, a ball is taken from the first bag. But if it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a white ball.

1. $\frac{1}{2}$
2. $\frac{14}{27}$
3. $\frac{4}{27}$
4. $\frac{16}{27}$

Q. A marksman is standing in front of a target participating in the regional championships. The probability of no foul happening is 75%. The chances that he hits the bullseye given there is no foul are 25%, while the chances of hitting bullseye with a foul are 15%. Find the probability of him hitting the bullseye.

1. 0.245
2. 0.352
3. 0.225
4. 0.255

Q. Three machines $E_1,E_2,E_3$ in a certain factory produced 50%, 25% and 25%, respectively, of the total daily output of electric tubes. It is known that 6% of the tubes produced on each of machines $E_1$ and $E_2$ is defective and that 5% of those produced on $E_3$ is defective. If one tube is picked up at random from a day’s production, calculate the probability that it is defective.

1. $\frac{3}{80}$
2. $\frac{9}{80}$
3. $\frac{1}{10}$
4. $\frac{23}{400}$

Q. Tweleve balls are distributed among three boxes. The probability that the first box contains three balls is

1. $\frac{110}{9}(\frac{2}{3}{)}^{10}$
2. $\frac{9}{110}(\frac{2}{3}{)}^{10}$
3. $\frac{{}^{12}{C}_3}{{12}^3}\times 2^9$
4. $\frac{{}^{12}{C}_3}{3^{12}}$

Q. If $A$ and $B$ are two events such that $P\left(A\right)=\frac{3}{4}$ and $P\left(B\right)=\frac{5}{8}$, then

1. $P(A\cup B)\ge \frac{3}{4}$
2. $P(A^{\prime }\cup B)\ge \frac{1}{4}$
3. $\frac{3}{8}\le P(A\cup B)\le \frac{5}{8}$
4. $\frac{3}{8}\le P(A\cap B)\le \frac{5}{8}$

Q. If the papers of $4$ students can be checked by any one of the $7$ teachers, then the probability that all the $4$ papers are checked by exactly $2$ teachers is

1. $\frac{2}{7}$
2. $\frac{12}{49}$
3. $\frac{32}{343}$
4. None of these

Q. Tweleve balls are distributed among three boxes. The probability that the first box contains three balls is

1. $\frac{110}{9}(\frac{2}{3}{)}^{10}$
2. $\frac{9}{110}(\frac{2}{3}{)}^{10}$
3. $\frac{{}^{12}{C}_3}{{12}^3}\times 2^9$
4. $\frac{{}^{12}{C}_3}{3^{12}}$

Q. An artillery target may be either at point I with probability $\frac{8}{9}$ or at point II with probability $\frac{1}{9}$. We have 55 shells, each of which can be fired either at point I or II. Each shell may hit the target, independent of the other shells, with probability $\frac{1}{2}$. Maximum number of shells that must be fired at point I to have maximum probability is

1. $20$
2. $25$
3. $29$
4. $35$

Q. A box $B_1$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B_2$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B_3$ contains 3 white ball, 4 red balls and 5 black balls. If one ball is drawn from each of the boxes $B_1,B_2$ and $B_3$, the probability that all three drawn balls are of the same colour is

1. $\frac{90}{648}$
2. $\frac{82}{648}$
3. $\frac{558}{648}$
4. $\frac{566}{648}$