Question Type 2: Finding the value of parameters that results in certain probabilities Exercises

A discrete random variable $X$ has $P(X=1)=k$, $P(X=2)=2k$, $P(X=3)=3k$ and $P(X=4)=1-6k$.

Find the set of possible values for $k$.

The density function of a nonnegative continuous variable is $f(x)=k e^{-x}$ for $x\ge0$. Find $k$.

A biased coin has probability of heads $k$. It is tossed twice. Given that $P(\text{no heads in two tosses})=0.16$, find $k$.

For events $A$ and $B$, $P(A)=0.5$, $P(B)=0.4$, and $P(A\cup B)=0.8$. Find the value of $P(A\cap B)$.

A fair six-sided die has events $A=\{\text{result}\le k\}$ and $B=\{\text{result is even}\}$. Find the values of the integer $k$, where $1 \le k \le 6$, such that $A$ and $B$ are independent.

X and Y are independent events with $P(X)=k$, $P(Y)=k-0.1$ and $P(X \cap Y)=k-0.18$. Find all possible values of $k$.

X and Y are independent with $P(X)=k$, $P(Y)=0.2$ and $P(X|Y)=0.5$. Find $k$.

Random variable $X$ is uniform on $[0,k]$. Find $k$ such that $P(X>1)=0.75$.

For two events $A$ and $B$, $P(A|B)=0.75$, $P(B)=0.4$ and $P(A)=k$.

Find the range of possible values for $k$.

Two independent events have $P(X)=2k$, $P(Y)=k+0.1$ and $P(X \cup Y)=0.8$. Find $k$.

A weighted die has $P(X=i)=k\,i$ for face $i=1,2,\dots,6$. Find $k$.