Yosuf has three grey socks and four blue socks in his sock drawer. He pulls out a pair of socks in the dark.
(a) What is the total number of possible outcomes?
(b) What is the probability that the pair of socks selected is grey?
(c) What is the probability that the pair of socks selected is blue?
(d) What is the probability that the pair of socks selected is neither grey nor blue?
a) 3 outcomes:
GG , GB, BB
b) 3 G, and 4B
prob (GG) = (3/7)(2/6) = 1/7
c) prob (BB) = ((4/7)(3/6) = 2/7
d) prob(not both blue or grey)
= prob (BG) + prob(GB)
= (4/7)(3/6) + (3/7)(4/6)
= 2/7 + 2/7 = 4/7
or we could have taken
1 - (1/7+2/7) = 4/7
To solve this problem, we can follow these steps:
Step 1: Determine the total number of possible outcomes.
Step 2: Find the number of favorable outcomes for each scenario.
Step 3: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes.
Now let's apply these steps to the given problem:
Step 1: Determine the total number of possible outcomes.
Yosuf has a total of 3 grey socks and 4 blue socks in his sock drawer. When he pulls out a pair of socks in the dark, there are several combinations possible. To find the total number of possible outcomes, we need to calculate the number of ways two socks can be chosen from the total number of socks. This can be done using the combination formula (nCr), where n is the total number of socks and r is the number of socks to be chosen.
In this case, n = 7 (3 grey + 4 blue socks), and r = 2 (since we are pulling out a pair of socks).
nCr = n! / (r! * (n-r)),
where "!" denotes factorial (the product of all positive integers less than or equal to n).
Using this formula, we can calculate the number of possible outcomes:
7C2 = 7! / (2! * (7-2)) = 21
So, there are a total of 21 possible outcomes.
(a) The total number of possible outcomes is 21.
Step 2: Find the number of favorable outcomes for each scenario.
For this step, we will calculate the number of outcomes that satisfy each condition (grey, blue, neither grey nor blue).
(b) What is the probability that the pair of socks selected is grey?
Since there are 3 grey socks, we need to find the number of ways to choose 2 grey socks from the available 3 grey socks. Using the combination formula:
3C2 = 3! / (2! * (3-2)) = 3
So, there are 3 favorable outcomes (pairs of grey socks).
(c) What is the probability that the pair of socks selected is blue?
Similarly, we can calculate the number of ways to choose 2 blue socks from the available 4 blue socks:
4C2 = 4! / (2! * (4-2)) = 6
So, there are 6 favorable outcomes (pairs of blue socks).
(d) What is the probability that the pair of socks selected is neither grey nor blue?
In this case, we need to calculate the number of ways to choose 2 socks from the remaining 7 - (3 + 4) = 0 "neither grey nor blue" socks. Mathematically, this is equal to:
0C2 = 0! / (2! * (0-2)) = 0
So, there are 0 favorable outcomes (pairs of neither grey nor blue socks).
Step 3: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes.
(b) The probability that the pair of socks selected is grey is:
3 (favorable outcomes) / 21 (total outcomes) = 1/7, approximately 0.143 or 14.3%.
(c) The probability that the pair of socks selected is blue is:
6 (favorable outcomes) / 21 (total outcomes) = 2/7, approximately 0.286 or 28.6%.
(d) The probability that the pair of socks selected is neither grey nor blue is:
0 (favorable outcomes) / 21 (total outcomes) = 0.
So, the probability is 0 (since there are no favorable outcomes for this scenario).