Question

Consider an experiment with a sample space S={car,train,unicycle, boat,bicycle,tricycle,motorcycle} . (Note: There is one of each vehicle.) What is the probability that an object has exactly 2 wheels, given that the object has wheels?

Answer 1

To determine the probability that an object has exactly 2 wheels given that it has wheels, we first identify the objects from the sample space \( S = \{ \text{car, train, unicycle, boat, bicycle, tricycle, motorcycle} \} \) that have wheels and then identify how many of those have exactly 2 wheels.

**Step 1: Identify objects with wheels.**
The vehicles in the sample space that have wheels are:

1. Unicycle (1 wheel)
2. Bicycle (2 wheels)
3. Tricycle (3 wheels)
4. Motorcycle (2 wheels)
5. Car (4 wheels)

**Total objects with wheels:**
- Unicycle
- Bicycle
- Tricycle
- Motorcycle
- Car

So, the objects with wheels are: **Unicycle, Bicycle, Tricycle, Motorcycle, Car**. This gives us a total of 5 objects with wheels.

**Step 2: Identify objects with exactly 2 wheels.**
From the identified objects, the ones that have exactly 2 wheels are:

1. Bicycle
2. Motorcycle

**Total objects with exactly 2 wheels:**
- Bicycle
- Motorcycle

So, there are **2 objects** with exactly 2 wheels.

**Step 3: Calculate the conditional probability.**
The probability that an object has exactly 2 wheels given that it has wheels can be calculated using the formula:

\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]

Where:
- \( A \) is the event that the object has exactly 2 wheels.
- \( B \) is the event that the object has wheels.

However, since we have already counted \( A \cap B \) and \( B \), we can simplify as follows:

\[
P(A | B) = \frac{\text{Number of favorable outcomes for A}}{\text{Total outcomes for B}} = \frac{2}{5}
\]

Thus, the probability that an object has exactly 2 wheels given that it has wheels is:

\[
\boxed{\frac{2}{5}}
\]