Let X and Y be two events such that P(X/Y) = 1/2, P(Y/X) = 1/3 and P(X ∩ Y) = 1/6. Which of the following is incorrect?

• P(X $\cup$ Y) = 2/3

• X and Y are independent

• P(X^C ∩ Y) = 2/3

• X and Y are not independent

If n integers taken at random are multiplied together, then the probability that the last digit of the product is 1, 3, 7 or 9 is

• $\frac{4^{\mathrm{n}}}{5^{\mathrm{n}}}$

• $\frac{2^{\mathrm{n}}}{5^{\mathrm{n}}}$

• 4 - $\frac{2^{\mathrm{n}}}{5^{\mathrm{n}}}$

• None

A bag contains (n + 1) coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is $\frac{7}{12}$, then the value of n is

• 5

• 4

• 3

• None of these

If A and B are two given events, then P(A ∩ B) is

• equal to P(A) + P(B)

• equal to P(A) + P(B) + P(A $\cup$ B)

• not less than P(A) + P(B) - 1

• not greater than P(A) + P(B) - P(A $\cup$ B)

From a city population, the probability of selecting a male or smoker $\frac{7}{10}$, a male smoker is $\frac{2}{5}$ and a male, if a smoker is already, selected, is $\frac{2}{3}$ Then, the probability of

• selecting a male is $\frac{3}{2}$

• selecting a smoker is $\frac{1}{5}$

• selecting a non-smoker is $\frac{2}{5}$

• selecting a smoker, if a male is first selected, is given by $\frac{8}{5}$

If A, B, C are three events associated with a random experiment, then $\mathrm{P}\left(\mathrm{A}\right)\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)\mathrm{P}\left(\frac{\mathrm{C}}{\mathrm{A}} \cap \mathrm{B}\right)$

• $\mathrm{P}\left(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}\right)$

• $\mathrm{P}\left(\mathrm{A}\hspace{0.17em}\cap \mathrm{B}\hspace{0.17em}\cap \mathrm{C}\right)$

• $\mathrm{P}\left(\frac{\mathrm{C}}{\mathrm{A}} \cap \mathrm{B}\right)$

• $\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)$

The probability of atleast one double six being thrown in n throws with two ordinary dice is greater than 99%.

Then, the least numerical value of n is

• 100

• 164

• 170

• 184

If the integers m and n are chosen at random from 1 to 100, then the probability that a number of the form 7ⁿ + 7^m is divisible by 5, equals to

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{1}{8}$

• $\frac{1}{3}$

Let X denote the sum of the numbers obtained when two fair dice are rolled. The variance and standard deviation of X are

• $\frac{31}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 31}}{{\displaystyle 6}}}$

• $\frac{35}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 35}}{{\displaystyle 6}}}$

• $\frac{17}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 17}}{{\displaystyle 6}}}$

• $\frac{31}{6} \mathrm{and} \sqrt{\frac{{\displaystyle 35}}{{\displaystyle 6}}}$

If two events A and B. If odds against A are 2 : 1 and those infavour of A $\cup$ B areas 3 : 1, then

• $\frac{1}{2} \le \mathrm{P}\left(\mathrm{B}\right) \le \frac{3}{4}$

• $\frac{5}{12} \le \mathrm{P}\left(\mathrm{B}\right) \le \frac{3}{4}$

• $\frac{1}{4} \le \mathrm{P}\left(\mathrm{B}\right) \le \frac{3}{5}$

• None of these