The question is incomplete! The complete question along with answers and explanation is provided below!
In your sock drawer you have 5 blue, 7 gray, and 2 black socks. Half asleep one morning you grab 2 socks at random and put them on. Find the probability you end up wearing the following socks. (Round your answers to four decimal places.)
a) 2 blue socks
b) no gray socks
c) at least 1 black sock
d) a green sock
e) matching socks
a) P(2 blue socks) = 0.1099 = 10.99%
b) P(no gray socks) = 0.2307 = 23.07%
c) P(at least 1 Black sock) = 0.2748 = 27.48%
d) P(green sock) = 0%
e) P(Matching socks) = 0.3516 = 35.16%
Step-by-step explanation:
Given Information:
5 Blue socks
7 Gray socks
2 black socks
Total socks = 5 + 7 + 2 = 14
a) The probability of wearing 2 blue socks
P(2 blue socks) = P(B1 and B2)
P(B1) = no. of blue socks/total no. of socks
P(B1) = 5/14 = 0.3571
Now there are 4 blue socks remaining and total 13 socks remaining
P(B2|B1) = 4/13 = 0.3077
P(B1 and B2) = 0.3571*0.3077 = 0.1099 = 10.99%
b) The probability of wearing no gray socks
5 Blue socks + 2 black socks = 7 socks are not gray
P(no gray socks) = P(Not G1 and Not G2)
P(Not G1) = no. socks that are not grey/ total no. of socks
P(Not G1) = 7/14 = 0.5
Now there are 6 socks remaining that are not gray and total 13 socks remaining
P(Not G2 | Not G1) = 6/13 = 0.4615
P(Not G1 and Not G2) = 0.5*0.4615 = 0.2307 = 23.07%
c) The probability of wearing at least 1 black sock
5 Blue socks + 7 Gray socks = 12 socks are not black
P(at least 1 Black) = 1 - P(Not B1 and Not B2)
P(Not B1) = no. socks that are not black/ total no. of socks
P(Not B1) = 12/14 = 0.8571
Now there are 11 socks remaining that are not black and total 13 socks remaining
P(Not B2 | Not B1) = 11/13 = 0.8461
P( Not B1 and Not B2) = 0.8571*0.8461 = 0.7252
P(at least 1 Black) = 1 - P( Not B1 and Not B2)
P(at least 1 Black) = 1 - 0.7252 = 0.2748 = 27.48%
d) The probability of wearing a green sock
There are 0 green socks, therefore
P(Green) = 0/14 = 0%
e) The probability of wearing matching socks
P(Matching socks) = P(2 Blue socks) + P(2 Gray socks) + P(2 Black socks)
P(2 Blue socks) already calculated in part a
P(2 Blue socks) = P(B1 and B2) = 0.1099
For Gray socks
P(G1) = no. of gray socks/ total no. of socks
P(G1) = 7/14 = 0.5
Now there are 6 gray socks remaining and total 13 socks remaining
P(G2 | G1) = 6/13 = 0.4615
P(2 Gray socks) = P(G1 and G2) = 0.5*0.4615 = 0.2307
For Black socks
P(B1) = no. of black socks/ total no. of socks
P(B1) = 2/14 = 0.1428
Now there is 1 black sock remaining and total 13 socks remaining
P(B2 | B1) = 1/13 = 0.0769
P(2 Black socks) = P(B1 and B2) = 0.1428*0.0769 = 0.0110
P(Matching socks) = P(2 Blue socks) + P(2 Gray socks) + P(2 Black socks)
P(Matching socks) = 0.1099 + 0.2307 + 0.0110 = 0.3516 = 35.16%
