a) Probability of ending up wearing 2 blue socks is 1/11.
b) Probability of ending up wearing no grey socks is 7/22.
c) Probability of ending up wearing at least 1 black sock is 5/11.
d) Probability of ending up wearing a green sock is 0.
e) Probability of ending up wearing matching socks is 19/66.
Step-by-step explanation:
Note: This question is not complete. The complete question is therefore provided before answering the question as follows:
In your sock drawer, you have 4 blue socks, 5 gray socks, and 3 black ones. Half asleep one morning, you grab 2 socks at random and put them on. Find the probability you end up wearing: a) 2 blue socks. b) no gray socks. c) at least 1 black sock. d) a green sock. e) matching socks.
The explanation of the answer is now given as follows:
The following are given in the question:
n(B) = number of Blue socks = 4
n(G) = number of Gray socks = 5
n(K) = number of black socks = 3
Therefore, we have:
n(T) = Total number of socks = n(B) + n(G) + n(K) = 4 + 5 + 3 = 12
To calculate a probability, the following formula for calculating probability is used:
Probability = Number of favorable outcomes / Number of total possible outcomes ……. (1)
Since this is a without replacement probability, we can now proceed as follows:
a) 2 blue socks
P(B) = Probability of ending up wearing 2 blue socks = ?
Probability of first pick = n(B) / n(T) = 4 / 12 = 1 / 3
Since it is without replacement, we have:
Probability of second pick = (n(B) – 1) / (n(T) – 1) = (4 – 1) / (12 – 1) = 3 / 11
P(B) = Probability of first pick * Probability of second pick = (1 / 3) * (3 / 11) = 1 / 11
b) no gray socks.
Number of favorable outcomes = n(B) + n(K) = 4 + 3 = 7
P(No G) = Probability of ending up wearing no gray socks = ?
Probability of first pick = Number of favorable outcomes / n(T) = 7 / 12
Since it is without replacement, we have:
Probability of second pick = (Number of favorable outcomes – 1) / (n(T) – 1) = (7 – 1) / (12 – 1) = 6 / 11
P(No G) = Probability of first pick * Probability of second pick = (7 / 12) * (6 / 11) = 7 / 22
c) at least 1 black sock.
Probability of at least one black sock = 1 - P(No K)
Number of favorable outcomes = n(B) + n(G) = 4 + 5 = 9
Probability of first pick = Number of favorable outcomes / n(T) = 9 / 12 = 3 /4
Since it is without replacement, we have:
Probability of second pick = (Number of favorable outcomes – 1) / (n(T) – 1) = (9 – 1) / (12 – 1) = 8 / 11
P(No K) = Probability of first pick * Probability of second pick = (3 / 4) * (8 / 11) = 24 / 44 = 6 / 11
Probability of at least one black sock = 1 - (6 / 11) = 5 / 11
d) a green sock.
n(Green) = number of Green socks = 0
Since, n(Green) = 0, it therefore implies that the probability of ending up wearing a green sock is 0.
e) matching socks.
This can be calculated using the following 4 steps:
Step 1: Calculation of the probability of matching blue socks
P(matching blue socks) = P(B) = 1 / 11
Step 2: Calculation of the probability of matching gray socks
P(matching green socks) = Probability of matching gray socks = ?
Probability of first pick = n(G) / n(T) = 5 / 12
Since it is without replacement, we have:
Probability of second pick = (n(G) – 1) / (n(T) – 1) = (5 – 1) / (12 – 1) = 4 / 11
P(matching gray socks = Probability of first pick * Probability of second pick = (5 / 12) * (4 / 11) = 20 / 132 = 5 / 33
Step 3: Calculation of the probability of matching black socks
P(matching black socks) = Probability of matching green socks = ?
Probability of first pick = n(K) / n(T) = 3 / 12 = 1 / 4
Since it is without replacement, we have:
Probability of second pick = (n(K) – 1) / (n(T) – 1) = (3 – 1) / (12 – 1) = 2 / 11
P(matching black socks) = Probability of first pick * Probability of second pick = (1 / 4) * (2 / 11) = 2 / 44 = 1 / 22
Step 4: Calculation of the probability of ending up wearing matching socks
P(matching socks) = Probability of ending up wearing matching socks = ?
P(matching socks) = P(matching blue socks) + P(matching grey socks) + P(matching black socks) = 1/11 + 5/33 + 1/22 = (6 + 10 + 3) / 66 = 19/66
