A closet contains 5 short-sleeve shirts and 3 long-sleeve shirts. Two shirts are chosen from the closet without looking. What is the probability that both shirts have short sleeves?
$\frac{25}{64}$
$\frac{2}{0}$
$\frac{9}{64}$
$\frac{3}{28}$
$\frac{5}{14}$
Answer (1)
Calculate the total number of ways to choose 2 shirts from 8: C ( 8 , 2 ) = 28 .
Calculate the number of ways to choose 2 short-sleeve shirts from 5: C ( 5 , 2 ) = 10 .
Divide the number of favorable outcomes by the total possible outcomes: 28 10 .
Simplify the fraction: 14 5 .
Explanation
Problem Analysis We have a closet with 5 short-sleeve shirts and 3 long-sleeve shirts. We want to find the probability of picking 2 short-sleeve shirts when we randomly select 2 shirts.
Total Possible Outcomes First, let's find the total number of ways to choose 2 shirts from the 8 shirts (5 short-sleeve + 3 long-sleeve). This is a combination problem, since the order in which we pick the shirts doesn't matter. The total number of combinations is given by: C ( n , k ) = k ! ( n − k )! n ! where n is the total number of items and k is the number of items to choose. In our case, n = 8 and k = 2 .
C ( 8 , 2 ) = 2 ! ( 8 − 2 )! 8 ! = 2 ! 6 ! 8 ! = 2 × 1 8 × 7 = 28 So there are 28 possible ways to choose 2 shirts from the closet.
Favorable Outcomes Next, let's find the number of ways to choose 2 short-sleeve shirts from the 5 short-sleeve shirts. Here, n = 5 and k = 2 .
C ( 5 , 2 ) = 2 ! ( 5 − 2 )! 5 ! = 2 ! 3 ! 5 ! = 2 × 1 5 × 4 = 10 So there are 10 ways to choose 2 short-sleeve shirts from the 5 short-sleeve shirts.
Calculate the Probability Now, we can calculate the probability of choosing 2 short-sleeve shirts by dividing the number of ways to choose 2 short-sleeve shirts by the total number of ways to choose 2 shirts: P ( 2 short-sleeve shirts ) = Total number of ways to choose 2 shirts Number of ways to choose 2 short-sleeve shirts = 28 10 We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2: P ( 2 short-sleeve shirts ) = 28 ÷ 2 10 ÷ 2 = 14 5
Final Answer Therefore, the probability that both shirts chosen are short-sleeve shirts is 14 5 .
Examples
This type of probability calculation is useful in many real-world scenarios, such as quality control in manufacturing. For example, if a factory produces a batch of items and you want to know the probability of selecting a certain number of defective items when you randomly pick a few for inspection, you would use combinations and probability calculations similar to this shirt problem. This helps ensure that the products meet the required quality standards. Another example is in card games, where you might want to calculate the probability of drawing specific cards from a deck.
