Respuesta :
Answer: The required expected value of X is 1.2.
Step-by-step explanation: Given that a jar contains 4 red marbles and 6 blue marbles. Someone reach in and chose two marbles at random.
We are to find the expected value of X, if X represents the number of blue marbles the person selected.
* X = 0, i.e., two red marbles and 0 blue marbles are selected.
The probability of selecting 0 blue marbles and 2 red marbles is
[tex]P(X=0)=\dfrac{^4C_2\times^6C_0}{^{10}C_2}=\dfrac{6\times1}{45}=\dfrac{2}{15}.[/tex]
* X = 1, i.e., one red marble and 1 blue marble is selected.
The probability of selecting 1 blue marble and 1 red marble is
[tex]P(X=1)=\dfrac{^4C_1\times^6C_1}{^{10}C_2}=\dfrac{4\times6}{45}=\dfrac{8}{15}.[/tex]
* X = 2, i.e., 0 red marbles and 2 blue marbles are selected.
The probability of selecting 2 blue marbles and 0 red marble is
[tex]P(X=2)=\dfrac{^4C_0\times^6C_2}{^{10}C_2}=\dfrac{1\times15}{45}=\dfrac{1}{3}.[/tex]
So, the probability distribution of X is given by
[tex]X~~~~~~~~~~~~~~~~~~~~0~~~~~~~~~~~~~~1~~~~~~~~~~~~~~~~~~2\\\\\\P(X)~~~~~~~~~~~~~~\dfrac{2}{15}~~~~~~~~~~~~\dfrac{8}{15}~~~~~~~~~~~~~~~~~\dfrac{1}{3}[/tex]
Therefore, the expected value of X is
[tex]E(X)=\sum XP(X)=0\times\dfrac{2}{15}+1\times\dfrac{8}{15}+2\times\dfrac{1}{3}=\dfrac{18}{15}=1.2.[/tex]
Thus, the required expected value of X is 1.2.
