Probability Tes...

Let A and B be two events such that $$P(A \cup B) \geq 3/4 $$ and $$1/8 \leq P( A \cap B) \leq 3/8$$.
Statement 1: $$P(A) + P(B) \geq 7/8$$
statement 2: $$P(A) + P(B) \leq 11/8$$

A biased coin with probability $$p,0<p<1$$, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $$2/5$$, then $$p=$$........

Events $$A,B,C$$ are mutually exclusive events such that $$\displaystyle P\left ( A \right )= \frac{3x+1}{3},$$ and $$\displaystyle P\left ( B \right )= \frac{1-x}{4},$$ and $$\displaystyle P\left ( C \right )= \frac{1-2x}{2}.$$ The set of possible values of $$x$$ is in the interval

The odds that a book will be reviewed favourably by three independent critics are 5 to 2,4 to 3 and 3 to 4 respectively; what is the probability that of three reviews a majority will be favourable?

Three numbers are chosen at random without replacement from $${1,2,....10}$$ . The probability that the minimum of the chosen numbers is $$3$$, or their maximum is $$7$$ is ..............

Let $$\displaystyle \omega$$ be a complex cube root of unity with $$\displaystyle \omega$$ $$\displaystyle \neq$$ 1. A fair die is thrown three times. If $$\displaystyle r_{1}, r_{2}$$and $$\displaystyle r_{3}$$ are the numbers obtained on the die then the probability that $$\displaystyle\omega  ^{r_{1}}+\omega ^{r_{2}}+\omega ^{r_{3}}=0$$ is

$$n$$ letters to each of which corresponds an addressed envelope are placed in the envelopes at random. What is the probability that no letter is placed in the right envelope?

A bag contains 4 white, 5 red and 6 black balls. Three are drawn at random. Find the probability that (i) no ball drawn is black, (ii) exactly 2 are black (iii) all are of the same colour.

If for two events $$A$$ and $$B$$,$$\displaystyle P\left ( A\cup B \right )=\frac{1}{2}, P\left ( A\cap B \right )=\frac{2}{5}$$and then $$P(A^{c})+P(B^{c})$$equals

$$A, B, C$$ are three events for which $$\displaystyle P(A) = 0.6, P(B) = 0.4, P(C) = 0.5, P(A \cup B) = 0.8, P(A \cap C) = 0.3$$ and $$\displaystyle P(A \cap B \cap C) = 0.2$$. If $$\displaystyle P(A \cup B \cup C) \geq 0.85$$ then the interval of values of $$\displaystyle P(B \cap C)$$ is