Two socks are put in a bag. There is equal probability that a sock that is put i

Question

Two socks are put in a bag. There is equal probability that a sock that is put into the bag is black and white. The process is that you draw socks out of the bag blindfolded and write the color down. The sock is returned back in the bag before you draw again.

a) If we draw a black sock two times in a row, what is the probabilty that both of the socks are black ?

b) If we draw a black sock two times in a row, what is the probabilty that we get a white sock in the third attempt

Answers are 2/3 and 1/6, but I cant figure out how to get that solution. Show calculations

Explanation / Answer

there are 3 possibilietes ; probability of 2 white socks in bag P(WW)=(1/2)*(1/2 )=1/4

probability of 2 black socks in bag =P(BB)=(1/2)*(1/2) =1/4

probability of 1 black sock and 1 white sock =P(WB) =1-1/4-1/4 =1/2

a) probability of 2 black sock =P(WW and 2 black socks+BB and 2 black socks+WB and 2 black socks)

=(1/4)*0*0+(1/4)*(1)*(1)+(1/2)*(1/2)*(1/2) =3/8

therefore probability of  that both of the socks are black given  black sock two times in a row

=P(BB and 2 black socks)/P(2 black socks)=(1/4)*(1)*(1)/(3/8) =2/3

b)

here probability of  that one black and one white socks given  black sock two times in a row

=P(WB and 2 black socks)/P(2 black socks)=(1/2)*(1/2)*(1/2)/(3/8) =1/3

therefore  probabilty that we get a white sock in the third attempt given 2 black socks two times in a row

=P(BB given two black socks two times in a row and white on 3rd attempt+WB given  two black socks two times in a row and white on 3rd attempt) =(2/3)*(0)+(1/3)*(1/2) =1/6

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