In: Statistics and Probability
You work at a sock factory where two varieties packs of socks are made. The first variety is simply 5 solid colors which are equally represented (20% of each color). The second variety is especially made for July 4th, it has 40% red socks, 30% blue socks, and 10% white, black, and grey socks. An accident happens and two boxes of socks are found with no labels, but we know that one box is the first variety and the other box is the second variety.
a) Is the first variety independent of the second variety?
b) You pull a sock out of the first box, what is the probability that the sock is red?
c)If the sock is red, what is the probability that box (first box) is the first variety? and what is the probability that the box is the second variety?
d)You pull out another sock from the first box, and it's a white sock. What is the probability that the first box is variety two? (the first and second sock you pulled are conditionally independent to each other with respects to the variety type. i.e. part d has nothing to do with part c)
e)You open the second box now and pull out a grey sock. Using all the given information what are the possibilities for each box being either the first or second varieties? (There should be 4 probabilities. Box one being the first variety, box one being the second variety, box two being the first variety, and box two being the second variety.)
a) The first variety is independent of the second variety
Becuase Pfirst variety is .5 = 1/2 and second variety also has probability 0.5 = 1/2
So P(A) +P(B) = 1 =.> so the 2 variaety are independent .
b) Given that the socks is pulled from the first pack There we have 5 diffenret colours of equal quantity
So P(Red) = 20% =0.2 or 1/5
c) it is a case of conditional probability Given that it is a red socks What is the probability that it is from Variaty 1 and Variety 2
We have the formula
B is occurance of red and A is probability of First variety
= 1/2x2/10=2/20=1/10=0.10
P(B)=P(of red )= out of 200% we have 20%+40% red = 60/200=30%=0.30
So P(A/B)= 0.10/0.30= 0.333
The second part is very similar
here = 1/2x4/10=4/20=1/5=0.20
So (P/B)= 0.20/0.30=0.667
d) We have given that pulled out socks is white and what is the probabilty that it us of variety 2
use the same formula
Here given value is that it is a white and B stands for white
A stands for variety 2
So = P( variety 2 and white )=1/2x1/10= 1/20=0.05
P(B) P(of white )= 20% from variety 1 and 10% from variety 2 So total 30/200=15/100=0.15
So P(A/B) = 0.05/0.15=5/15=.333
e) WE are pullling out a grey socks
Probability of first box variety 1
prob first box variety 2
Second box variety 1
Second box variety 2
P(Box 1 being 1 st) = P (Box 2 being second )= Given condition is a grey socks
Pro second being variety 2= 1/2= P(A)
P(grey) = P(B)= 30/200=.15
= Grey in variety 2 =0.10
So the Probability is = 1/2x 0.10/0.15=1/2x2/3=2/6=1/3=.333
Similarly the P(Box 2 neing 1st = P(1 being second)= 1/2x0.1./0.20=1/2x1/2=1/4=0.25