mike has 4 different-colored bath towels and 3 different-colored hand towels in his closet. only one of the bath towels is the same color as one of the hand towels.

Question

mike randomly takes one bath towel and one hand towel from the closet.

what is the probability that mike will select the two towels of the same color?

Answer 1

To determine the probability that Mike selects two towels of the same color, let's break down the problem step by step.

1. Identification of Towels:

   • Mike has 4 different-colored bath towels and 3 different-colored hand towels.
   • Among these, only one of the bath towels is the same color as one of the hand towels. Let's denote the colors as follows:
     • Bath Towels: \( B_1, B_2, B_3, B_4 \) (where \( B_1 \) is the towel that matches a hand towel)
     • Hand Towels: \( H_1, H_2, H_3 \) (where \( H_1 \) is the one that matches \( B_1 \))

2. Possible Selections:

   • When Mike selects one bath towel and one hand towel, he has:
     • 4 choices for the bath towel (i.e., \( B_1, B_2, B_3, B_4 \))
     • 3 choices for the hand towel (i.e., \( H_1, H_2, H_3 \))

3. Total Outcomes:

   • The total number of combinations of bath towels and hand towels selected is: \[ 4 \text{ (choices for bath towels)} \times 3 \text{ (choices for hand towels)} = 12 \text{ total outcomes} \]

4. Favorable Outcomes:

   • The only pair of towels that have the same color is when Mike picks the matching bath towel \( B_1 \) and the matching hand towel \( H_1 \). This counts as 1 favorable outcome.

5. Calculating Probability:

   • The probability \( P \) that Mike selects a bath towel and a hand towel of the same color is calculated by taking the ratio of favorable outcomes to total outcomes: \[ P = \frac{\text{number of favorable outcomes}}{\text{total outcomes}} = \frac{1}{12} \]

Thus, the probability that Mike will select two towels of the same color is \(\frac{1}{12}\).