Question

Jonathan is getting dressed in the dark. He has three drawers of socks. The top drawer contains 5 blue and 5 red

socks, the middle drawer contains 6 blue and 4 red socks, and the bottom drawer contains 3 blue and 2 red socks.
Jonathan will open one drawer and will select two socks at random.
a. Which drawer should he choose in order to make it most likely that he will select 2 red socks?
b. Which drawer should he choose in order to make it most likely that he will select 2 blue socks?
c. Which drawer should he choose in order to make it most likely that he will select a matching pair?

Answer 1

a. It is most likely that he will select 2 red socks if he chooses the top drawer.

b. It is most likely that he will select 2 blue socks if he chooses the middle drawer.

c. It is most likely that he will select a matching pair if he chooses the middle drawer.

Explanation:

This is a case of choosing elements with replacement, this means that if you choose an element from the set, it will be out of the set and the total amount of data will be reduced. In the following explanation Drawer 1 will be the top one, Drawer 2 the middle one and Drawer 3 the bottom one.

a. Which drawer should he choose in order to make it most likely that he will select 2 red socks?

To find this probability, you have to find the probability to choose a red sock the first time and a red one the second time too. Then you multiply these two probabilities and this is the total probability. For each drawer is as follows:

Drawer 1:

P1(red)=5/10

P2(red)=4/9

P(red,red)=(5/10)*(4/9)=2/9=0.22

Drawer 2:

P1(red)=4/10

P2(red)=3/9

P(red,red)=(4/10)*(3/9)=2/15=0.13

Drawer 3:

P1(red)=2/5

P2(red)=1/4

P(red,red)=(2/5)*(3/5)=1/10=0.1

P(red,red) is greater for Drawer 1.

b. Which drawer should he choose in order to make it most likely that he will select 2 blue socks?

To find this probability, you have to find the probability to choose a blue sock the first time and a blue one the second time too. Then you multiply these two probabilities and this is the total probability. For each drawer is as follows:

Drawer 1:

P1(blue)=5/10

P2(blue)=4/9

P(blue,blue)=(5/10)*(4/9)=2/9=0.22

Drawer 2:

P1(blue)=6/10

P2(blue)=5/9

P(blue,blue)=(6/10)*(5/9)=1/3=0.33

Drawer 3:

P1(blue)=3/5

P2(blue)=2/4

P(blue,blue)=(3/5)*(2/5)=3/10=0.3

P(blue,blue) is greater for Drawer 2.

c. Which drawer should he choose in order to make it most likely that he will select a matching pair?

In this case, you can choose two red socks or two blue socks. You can find this probability by adding P(blue,blue)+P(red,red) for each drawer.

Drawer 1:

2/9+2/9=4/9=0.44

Drawer 2:

2/15+1/3=7/15=0.46

Drawer 3:

1/10+3/10=4/10=0.4

The probability is grater for Drawer 2.