Suppose your top drawer contains different colored socks: 10 are white, 8 are black, 14 are pink, and 12 are blue. All socks in the drawer are loose (unpaired). In the morning, you randomly select two socks, one at a time.
What is the probability that you get either a white pair or a blue pair of socks?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
P(w) = 10/44 * (10-1)/(44-1) = ?
P(b) = 12/44 * (12-1)/(44-1) = ?
Either-or probabilities are found by adding the individual probabilities.
P(w) + P(w) = ?
To find the probability of selecting either a white pair or a blue pair of socks, we first need to determine the total number of possible pairs we can form from the given colors.
Step 1: Calculate the total number of socks.
In this case, we have 10 white socks, 8 black socks, 14 pink socks, and 12 blue socks. So the total number of socks in the drawer is 10 + 8 + 14 + 12 = 44 socks.
Step 2: Calculate the total number of possible pairs.
To form a pair, we need to select two socks. Since the socks are loose and unpaired, the order in which we select them does not matter. Therefore, we can use the combination formula to find the total number of possible pairs. The formula is:
nCr = n! / (r! * (n - r)!)
For this problem, we have:
n = 44 (total number of socks)
r = 2 (number of socks we want to select for a pair)
Using the combination formula, we can calculate:
44C2 = 44! / (2! * (44 - 2)!)
Simplifying the formula gives us:
44C2 = 44! / (2! * 42!)
44C2 = (44 * 43) / (2 * 1)
44C2 = 946
So, there are 946 possible pairs of socks we can form from the drawer.
Step 3: Calculate the number of pairs that are either white or blue.
For a pair to be either white or blue, both socks in the pair must be of the same color. Therefore, we need to calculate the number of white pairs and the number of blue pairs.
Number of white pairs:
To form a white pair, we need to select two white socks. Using the combination formula:
10C2 = 10! / (2! * (10 - 2)!)
10C2 = (10 * 9) / (2 * 1)
10C2 = 45
Number of blue pairs:
To form a blue pair, we need to select two blue socks. Using the combination formula:
12C2 = 12! / (2! * (12 - 2)!)
12C2 = (12 * 11) / (2 * 1)
12C2 = 66
Adding the number of white pairs and blue pairs gives us:
Number of white pairs + Number of blue pairs = 45 + 66 = 111 pairs
Step 4: Calculate the probability of selecting either a white pair or a blue pair.
The probability is given by the formula:
Probability = Number of successful outcomes / Total number of possible outcomes
In this case, the number of successful outcomes is the number of pairs that are either white or blue, which is 111, and the total number of possible outcomes is 946.
Probability = 111 / 946
Using a calculator to divide 111 by 946 gives us:
Probability ≈ 0.1174
Therefore, the probability of selecting either a white pair or a blue pair of socks is approximately 0.1174, which is equivalent to around 11.74%.