Question

A package of balloons contains 4 red, 6 blue, 3 yellow, and 5 orange balloons. Javier will randomly choose one balloon from the package to blow up. He then will randomly choose a second balloon to blow up.

What is the probability that Javier chooses an orange balloon and then a yellow balloon?

Answer options with 5 options
A.
StartFraction 5 over 108 EndFraction

B.
StartFraction 5 over 102 EndFraction

C.
StartFraction 1 over 15 EndFraction

D.
StartFraction 1 over 6 EndFraction

E.
StartFraction 4 over 9 EndFraction

Answer 1

To find the probability that Javier chooses an orange balloon and then a yellow balloon, we start by identifying the total number of balloons in the package.

The number of balloons is calculated as follows:
- Red: 4
- Blue: 6
- Yellow: 3
- Orange: 5

Total number of balloons = \(4 + 6 + 3 + 5 = 18\)

Now we will find the probability of the two events happening in sequence:

1. **Probability of choosing an orange balloon first:**
The probability of choosing an orange balloon is the number of orange balloons divided by the total number of balloons:
\[
P(\text{Orange first}) = \frac{5}{18}
\]

2. **Probability of choosing a yellow balloon second (after having already chosen an orange balloon):**
After Javier has chosen one orange balloon, there will be 17 balloons remaining (18 - 1 = 17). The number of yellow balloons remains the same (3).
The probability of choosing a yellow balloon after choosing an orange balloon is:
\[
P(\text{Yellow second | orange first}) = \frac{3}{17}
\]

3. **Combined probability of both events:**
To find the combined probability of both events happening, we multiply the probabilities of the two independent events:
\[
P(\text{Orange first and Yellow second}) = P(\text{Orange first}) \times P(\text{Yellow second | orange first}) = \frac{5}{18} \times \frac{3}{17} = \frac{15}{306}
\]

4. **Simplifying the result:**
To simplify \(\frac{15}{306}\), we can divide both the numerator and the denominator by 3:
\[
\frac{15 \div 3}{306 \div 3} = \frac{5}{102}
\]

Thus, the probability that Javier chooses an orange balloon and then a yellow balloon is:

\[
\boxed{\frac{5}{102}}
\]

The correct answer is option B: \(\frac{5}{102}\).