Answer:
E (X) = 1.8
Var (X) = 0.36
σ = 0.6
Step-by-step explanation:
Solution:-
- Denote the random variable X : is the number of red marbles that Suzan has in her hand after she selects three marbles.
- Total sample space (bag) have the following quantity of colored marbles:
Bag : { 3 Red , 2 Green }
- Suzan selects three marbles from the bag. The Event (X) defines the number of red marbles out of 3.
- The total number of outcomes / selections for randomly selecting 3 balls from the bag:
All outcomes = 5 C 3 = 10
- The probability distribution of the random variable X, we will use combinations to determine the required probabilities:
X = 1 red marble:
P ( X = 1 ) : Suzan chooses 1 Red marble from the available 3 red marble and 2 green marbles.
P ( X = 1 ) = [ 3C1*2C2 ] / all outcomes = (3*1) / 10 = 0.3
X = 2 red marble:
P ( X = 2 ) : Suzan chooses 2 Red marble from the available 3 red marble and 1 green marbles.
P ( X = 2 ) = [ 3C2*2C1 ] / all outcomes = (3*2) / 10 = 0.6
X = 3 red marble:
P ( X = 3 ) : Suzan chooses 3 Red marble from the available 3 red marble.
P ( X = 3 ) = [ 3C3] / all outcomes = (1) / 10 = 0.1
- The probability distribution is as follows:
X : 1 2 3
P (X): 0.3 0.6 0.1
- The expected value E(X) for the given random variable X is:
E ( X ) = ∑Xi*P(Xi)
= 1*0.3 + 2*0.6 + 3*0.1
=1.8
- The variance Var(X) for the given random variable X is:
Var ( X ) = ∑Xi^2*P(Xi) - [ E(X) ] ^2
= 1^2*0.3 + 2^2*0.6 + 3^2*0.1 - 1.8^2
= 0.36
- The standard deviation for the given random variable X is:
σ = √Var(X)
σ = √0.36
σ = 0.6
is 5 times
, which means
, so we have
. This means that
So, substituting this back gives