Respuesta :
Answer:
(a) [tex]\frac{3}{44}[/tex]
(b) [tex]\frac{1}{22}[/tex]
(c) [tex]\frac{5}{22}[/tex]
Step-by-step explanation:
We are given that Two marbles are drawn randomly one after the other without replacement from a jar that contains 3 red marbles, 3 white marbles, and 6 yellow marbles.
So, total number of marbles = 3 + 3 + 6 = 12 marbles
(a) Probability that a red marble is drawn first followed by a white marble is given by;
First P(drawing red marble) = [tex]\frac{3}{12}[/tex]
Now, since one red marble has been drawn so now the total marbles remaining will be 12 - 1 = 11 marbles and from this white marble will be drawn.
P(drawing white marble) = [tex]\frac{3}{11}[/tex]
Therefore, Probability that a red marble is drawn first followed by a white marble = [tex]\frac{3}{12} * \frac{3}{11}[/tex] = [tex]\frac{3}{44}[/tex] .
(b) Probability that a white marble is drawn first followed by a white marble is given by;
P(drawing first white marble) = [tex]\frac{3}{12}[/tex]
Now, the white marbles remaining are 3 - 1 = 2 marbles and total number of marbles are 12 - 1 = 11 marbles.
So, P(drawing another white marble) = [tex]\frac{2}{11}[/tex]
Therefore, Probability that a white marble is drawn first followed by a white marble = [tex]\frac{3}{12} * \frac{2}{11}[/tex] = [tex]\frac{1}{22}[/tex] .
(c) Probability that a yellow marble is not drawn at all is given by;
Now both the marbles will be drawn from red and white marbles i.e. 6 marbles and total marbles will be 12
So, probability that a yellow marble is not drawn at all = [tex]\frac{^{6}C_2 }{^{12}C_2}[/tex] = [tex]\frac{\frac{6!}{2!*4!} }{\frac{12!}{2!*10!} }[/tex]
= [tex]\frac{6!}{4!} * \frac{10!}{12!}[/tex] = [tex]\frac{6*5*4!}{4!} * \frac{10!}{12*11*10!}[/tex] = [tex]\frac{30}{132}[/tex] = [tex]\frac{5}{22}[/tex] .
