Answer:
Then the probability distribution is:
P(0) = 1/3
P(1) = 4/15
P(2) = 1/5
P(3) = 2/15
P(4) = 1/15
The expected value for X is:
EV = 1.33...
Step-by-step explanation:
We have a total of 6 marbles in the jar.
The probability of getting a red marble in the first try (X = 0) is equal to the quotient between the number of red marbles and the total number of marbles, this is:
P(0) = 2/6 = 1/3
The probability of drawing one green marble (X = 1)
is:
First, you draw a green marble with a probability of 4/6
Then you draw the red one, but now there are 5 marbles in the jar (2 red ones and 3 green ones), then the probability is 2/5
The joint probability is:
P(1) = (4/6)*(2/5) = (2/3)*(2/5) = 4/15
The probability of drawing two green marbles (X = 2)
Again, first we draw a green marble with a probability of 4/6
Now we draw again a green marble, now there are 3 green marbles and 5 total marbles in the jar, so this time the probability is 3/5
Now we draw the red marble (there are 2 red marbles and 4 total marbles in the jar), with a probability of 2/4
The joint probability is:
P(2) = (4/6)*(3/5)*(2/4) = (2/6)*(3/5) = 1/5
The probability of drawing 3 green marbles (X = 3)
At this point you may already understand the pattern:
First, we draw a green marble with a probability 4/6
second, we draw a green marble with a probability 3/5
third, we draw a green marble with a probability 2/4
finally, we draw a red marble with a probability 2/3
The joint probability is:
P(3) = (4/6)*(3/5)*(2/4)*(2/3) = (2/6)*(3/5)*(2/3) = (1/5)*(2/3) = (2/15)
Finally, the probability of drawing four green marbles (X = 4) is given by:
First, we draw a green marble with a probability 4/6
second, we draw a green marble with a probability 3/5
third, we draw a green marble with a probability 2/4
fourth, we draw a green marble with a probability 1/3
Finally, we draw a red marble with a probability 2/2 = 1
The joint probability is:
P(4) = (4/6)*(3/5)*(2/4)*(1/3)*1 = (1/5)*(1/3) = 1/15
Then the probability distribution is:
P(0) = 1/3
P(1) = 4/15
P(2) = 1/5
P(3) = 2/15
P(4) = 1/15
The expected value will be:
EV = 0*P(0) + 1*P(1) + 2*P(2) + 3*P(3) + 4*P(4)
EV = 1*(4/15) + 2*( 1/5) + 3*( 2/15) + 4*(1/15 ) = 1.33
So we can expect to draw 1.33 green marbles in this experiment.
