A fair six-faced die is rolled 12 times. The probability that each face turns up twice is equal to

• $\frac{12!}{6! 6! 6^{12}}$

• $\frac{2^{12}}{2^6{6}^{12}}$

• $\frac{12!}{2^6{6}^{12}}$

• $\frac{12!}{6^2{6}^{12}}$

A poker hand consists of 5 cards drawn at random from a well-shuffled pack of 52 cards. Then, the probability that a poker hand consists of a pair and a triple of equal face values (for example, 2 sevens and 3 kings or 2 aces and 3 queens, etc.) is

• $\frac{6}{4165}$

• $\frac{23}{4165}$

• $\frac{1797}{4165}$

• $\frac{1}{4165}$

Each of a and b can take values 1 or 2 with equal probability. The probability that the equation ax² + bx + 1= 0 has real roots, is equal to

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{1}{16}$

Cards are drawn one-by-one without replacement from a well shuffled pack of 52 cards. Then, the probability that a face card (jack, queen or king) will appear for the first time on the third turn is equal to

• $\frac{300}{2197}$

• $\frac{36}{85}$

• $\frac{12}{85}$

• $\frac{4}{51}$

There are two coins, one unbiased with probaility $\frac{1}{2}$ or getting heads and the other one is biased with probability $\frac{3}{4}$ of getting heads. A coin is selected at random and tossed. It shows heads up. Then, the probability that the unbiased coin was selected is

• $\frac{2}{3}$

• $\frac{3}{5}$

• $\frac{1}{2}$

• $\frac{2}{5}$

Two decks of playing cards are well shuffled and 26 cards are randomly distributed to a player. Then, the probability that the player gets all distinct cards is

• ${}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

• $2 \times {}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

• $2^{13} \times {}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

• $2^{26} \times {}^{52}\mathrm{C}_{\mathrm{r}}/{}^{104}\mathrm{C}_{26}$

An um contains 8 red and 5 white balls. Three balls are drawn at random. Then, the probability that balls of both colours are drawn is

• $\frac{40}{143}$

• $\frac{70}{143}$

• $\frac{3}{13}$

• $\frac{10}{13}$

Let A and B be two events with $\mathrm{P}\left({\mathrm{A}}^{\mathrm{C}}\right) = 0.3, \mathrm{P}\left(\mathrm{B}\right) = 0.4 \mathrm{and} \mathrm{P}\left(\mathrm{A} \cap {\mathrm{B}}^{\mathrm{C}}\right)$. Then, $\mathrm{P}\left(\mathrm{B} | \mathrm{A} \cup {\mathrm{B}}^{\mathrm{C}}\right)$ is equal to

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

29.

4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{6}$

A coin is tossed again and again. If tail appears on first three tosses, then the chance that head appears on fourth toss is

• $\frac{1}{16}$

• $\frac{1}{2}$

• $\frac{1}{8}$

• $\frac{1}{4}$