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:
-
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: \[ P(\text{Orange first}) = \frac{5}{18} \]
-
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: \[ P(\text{Yellow second | orange first}) = \frac{3}{17} \]
-
Combined probability of both events: To find the combined probability of both events happening, we multiply the probabilities of the two independent events: \[ P(\text{Orange first and Yellow second}) = P(\text{Orange first}) \times P(\text{Yellow second | orange first}) = \frac{5}{18} \times \frac{3}{17} = \frac{15}{306} \]
-
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: \(\frac{5}{102}\).
1 answer