Let's break down the problem step-by-step to find the probability of selecting a blue piece of clothing or a pair of pants from the given box.
1. Determine the Total Number of Pieces of Clothing:
- Number of black shirts: 4
- Number of blue shirts: 8
- Number of black pants: 4
- Number of blue pants: 10
To find the total number of pieces of clothing, we add all these quantities together:
[tex]\[
\text{Total pieces of clothing} = 4 + 8 + 4 + 10 = 26
\][/tex]
Hence, there are 26 pieces of clothing in total.
2. Calculate the Probability of Selecting a Blue Piece of Clothing (Event A):
- Number of blue shirts: 8
- Number of blue pants: 10
The total number of blue pieces of clothing is:
[tex]\[
\text{Total blue clothing} = 8 + 10 = 18
\][/tex]
The probability of selecting a blue piece of clothing is:
[tex]\[
P(\text{Blue clothing}) = \frac{\text{Total blue clothing}}{\text{Total pieces of clothing}} = \frac{18}{26} \approx 0.6923
\][/tex]
3. Calculate the Probability of Selecting a Pair of Pants (Event B):
- Number of black pants: 4
- Number of blue pants: 10
The total number of pants is:
[tex]\[
\text{Total pants} = 4 + 10 = 14
\][/tex]
The probability of selecting a pair of pants is:
[tex]\[
P(\text{Pants}) = \frac{\text{Total pants}}{\text{Total pieces of clothing}} = \frac{14}{26} \approx 0.5385
\][/tex]
4. Calculate the Probability of Selecting a Blue Pair of Pants (Event A and Event B together):
- Number of blue pants: 10
The probability of selecting a blue pair of pants is:
[tex]\[
P(\text{Blue pants}) = \frac{10}{26} \approx 0.3846
\][/tex]
5. Calculate the Probability of Selecting a Blue Piece of Clothing or a Pair of Pants:
We need to use the formula for the union of two events:
[tex]\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\][/tex]
Substituting the values we have calculated:
[tex]\[
P(\text{Blue clothing or Pants}) = P(\text{Blue clothing}) + P(\text{Pants}) - P(\text{Blue pants})
\][/tex]
[tex]\[
P(\text{Blue clothing or Pants}) = 0.6923 + 0.5385 - 0.3846 = 0.8462
\][/tex]
Therefore, the probability of randomly selecting a blue piece of clothing or a pair of pants is approximately [tex]\(0.8462\)[/tex] or [tex]\(84.62\%\)[/tex].
