$A$ coin is tossed three times. If $X$ denotes the absolute difference between the number of heads and the number of tails,then $P(X=1) = $

In a game,$3$ coins are tossed. $A$ person is paid ₹ $7$ if he gets all heads or all tails,and he is supposed to pay ₹ $3$ if he gets one head or two heads. The amount he can expect to win on an average per game is ₹

$A$ random variable $X$ has the range $\{0, 1, 2, \ldots\}$. If $P(X=r) = k(1+r) 3^{-r}$ for $r=0, 1, 2, \ldots$, where $k > 0$ is a real number, then $P(X=0) + P(X=1) + P(X=2) =$

$A$ random variable $X$ has the following probability distribution:
$X = 1, 2, 3, 4, 5$
$P(X) = 0.1, 0.2, 0.3, 0.2, 0.2$
For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 4\}$, find $P(E \cup F)$.

$A$ random variable $X$ has the following probability distribution:
| $x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
|---|---|---|---|---|---|---|---|---|
| $P(x)$ | $0.15$ | $0.23$ | $0.12$ | $0.10$ | $0.20$ | $0.08$ | $0.07$ | $0.05$ |
For the events $E = \{x \text{ is a prime number}\}$ and $F = \{x < 4\}$,the probability $P(E \cup F)$ is:

If the $c.d.f.$ (cumulative distribution function) is given by $F(x) = \frac{x-25}{10}$,then $P(27 \leq x \leq 33) = \_\_\_\_$