There are four brown shoes and 10 black shoes on the floor. your puppy cares away two shoes and puts one shoe in the trashcan and one in the laundry basket.
a. What is the probability that there will be a black shoe in the trash and a brown shoe in the laundry basket?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
4/14 * 10/13 = 10/14 * 4/13 = ?
The total is reduced to 13, because one shoe has already been taken.
To find the probability, we can consider the number of favorable outcomes and the total number of possible outcomes.
In this case, there are 4 brown shoes and 10 black shoes initially on the floor. From these, the puppy takes away two shoes randomly and places one in the trash and one in the laundry basket.
The total number of possible outcomes can be calculated by considering the choices the puppy made for each shoe. Since two shoes were chosen from 14 shoes (4 brown + 10 black), the total number of possible outcomes is given by the combination:
Total number of outcomes = C(14, 2) = 14! / (2!(14-2)!) = 91
Now, let's consider the favorable outcomes. We want one black shoe in the trash and one brown shoe in the laundry basket.
The number of favorable outcomes can be calculated by considering the choices the puppy made for the black shoe and the brown shoe. Since the puppy took away two shoes randomly, the number of favorable outcomes is given by the product:
Number of favorable outcomes = C(10, 1) * C(4, 1) = (10! / (1!(10-1)!)) * (4! / (1!(4-1)!)) = 10 * 4 = 40
Therefore, the probability that there will be a black shoe in the trash and a brown shoe in the laundry basket is:
Probability = Number of favorable outcomes / Total number of outcomes = 40 / 91 ≈ 0.4396 (rounded to 4 decimal places)
So, the probability is approximately 0.4396.
To determine the probability of having a black shoe in the trash and a brown shoe in the laundry basket, we first need to calculate the total number of shoes remaining after the puppy's actions.
Initially, there were 4 brown shoes and 10 black shoes on the floor, making a total of 14 shoes.
However, the puppy carried away two shoes. Therefore, there will be 14 - 2 = 12 shoes remaining.
Now we need to consider the possibilities of what the puppy did with the shoes. There are two scenarios to consider:
1. The puppy put a black shoe in the trash and a brown shoe in the laundry basket.
- The probability of the puppy picking a black shoe for the trash is 10 black shoes out of 12 remaining shoes, which is 10/12.
- The probability of the puppy picking a brown shoe for the laundry basket is 4 brown shoes out of the remaining 11 shoes (since one shoe is already in the trash), which is 4/11.
- To calculate the probability of both events occurring together, we multiply the individual probabilities: (10/12) * (4/11).
Therefore, the probability of having a black shoe in the trash and a brown shoe in the laundry basket is: (10/12) * (4/11) = 40/132, which simplifies to 10/33.