Question

If you randomly draw 3 marbels from a bag that contains 3 red and 5 green marbles, without replacement. Denote by X the number of red marbels drawn.
(a) What is the probability that exactly 1 red marbel is drawn?
(b) Find the probability distribution of X.
(c) FInd the expected value and standard deviation of X.
(d) Draw the probability distribution of X and mark where the expected value is and one standard deviation away from the mean in both the positive and negative direction.

Answer 1

a) .

There are 3 red marbles and 5 green marbles. Also, the sample is drawn without replacement.

Hence, Number of ways of selecting exactly 1 red marble in the sample of 3 marbles= 3C1 * 5C2

Total number of ways of selecting 3 marbles from total 8 marbles = 8C3

Hence, the probability of selecting exactly 1 red marble is drawn = ( 3C1 * 5C2 ) / 8C3

= (3*10) / 56

= 0.5357 is the required probabilty

Suppose, the bag contains N marbles of which M red, (N-M) green marbles. A sample of 'n' marbles is drawn without replacement from N marbles. X denote the number of red marbles (in a randomly drawn sample)

So, X can take valoes 0,1,2,....n (assuming n<=M)

The probabilty distribution is obtained as,

We want P(X=x),

If the sample of n marbles contains 'x' red marbles, then it will contain 'n-x' green marbles. Hence, the number of ways in which x red marbles can be selected from M red marbles is MCx.

and the number of ways in which n-x green marbles can be selected from N-M black marbles is N-MCn-x. The sample contains both red and green marbles. Therefore, the total number of ways

in which the above event can occur is  MCx *  N-MCn-x . In all 'n'  marbles are selected from N marbles. Therefore, total numer of possible ways of selection is NCn.

Hence, the probabilty of this is,

The above P(x) is called as the probability mass function of HYPEGEOMETIC DISTIBUTION  with parameters (N,M,n-m)   

Denoted as H(N,M,n)

( Here, if we donot assume n<=M, then the range 0f X is 0,1,2,.... min(n,M). Because at most M red marbles can be there in the sample.)

c) The mean and variance of X which follows Hypergeometric distribution is

Here, n=3, M= 3, N= 8

Therefore Standard deviation is,

d)  

The Probability distribution of X is plotted using R as ,

> x=0:3
> y=dhyper(0:3, 3, 5, 3, log = FALSE)
> plot(x,y,type="h",lwd=2, xlab='x',ylab='f(x)=P(x)',main='Probability distribution of x')