Probability Questions

Suppose $P(A)=0.4$, $P(B)=P$, and $P(A\cup B)=0.7$. If $A$ and $B$ are independent events, then the value of $P$ is:

If four dice are thrown together, what is the probability that the sum of the numbers appearing on them is 25?

The chances of a defective screw in three boxes A, B, and C are $1/5$, $1/6$, and $1/7$ respectively. A box is selected at random and a screw is found to be defective. Find the probability that it came from box A.

If $P(A)=0.4$, $P(B)=0.8$, and $P(B|A)=0.6$, then $P(A\cup B)$ is equal to:

$A$ and $B$ are two independent witnesses in a case. The probability that $A$ will speak the truth is $1/3$ and the probability that $B$ will speak the truth is $3/4$. If $A$ and $B$ agree on a certain statement, then the probability that the statement is true is:

For any two events $A$ and $B$, $P(A)=P(A|B)=1/4$ and $P(B|A)=1/2$. Then which of the following is NOT correct?

A fair die is thrown six times. The probability of getting a factor of 6 exactly four times is:

Below is a figure indicating a sample space $S$ and the probability of occurrences of events $A$ and $B$. Which colored section of the portion will indicate the probability of events $B$ and $A$ occurring simultaneously?

A continuous random variable $x$ has the probability density function:$f(x)=\{\begin{array}{ll}{e}^x,& -1\le x<0\\ \frac{1}{1+x^2},& 0\le x<1\\ \text{elsewhere},& e-1-\frac{\pi }{4}\end{array}$$P(-1<x<1)$ is equal to:

If the probability of $A$ to fail in an examination is $0.2$ and that for $B$ is $0.3$, then the probability that either $A$ or $B$ fails (meaning exactly one of them fails) is:

A coin is tossed $3$ times. If $X$ is a random variable representing the number of tails, then the mathematical expectation of $X$ is: