Your science teacher sets up six flasks. Two of the flasks contain water and four of the flasks contain hydrogen peroxide. A reaction occurs when you add yeast to hydrogen peroxide. You add yeast to two of the flasks. What is the probability that at least one reaction will occur?
The Correct Answer and Explanation is :
To solve this problem, let’s break it down step by step.
Given:
- There are 6 flasks in total: 2 contain water, and 4 contain hydrogen peroxide.
- You add yeast to 2 of the flasks.
- A reaction occurs only when yeast is added to a flask containing hydrogen peroxide.
Goal:
We need to calculate the probability that at least one reaction occurs. In other words, we want to find the probability that at least one of the two flasks you add yeast to contains hydrogen peroxide.
Step 1: Total Number of Ways to Choose 2 Flasks
There are 6 flasks in total, and you need to select 2 to add the yeast. The total number of ways to choose 2 flasks from 6 is given by the combination formula:
[
\text{Total ways} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15
]
So, there are 15 possible ways to choose 2 flasks.
Step 2: Calculate the Number of Unfavorable Outcomes
The unfavorable outcomes are the cases where neither of the two flasks you select contains hydrogen peroxide. If you choose from the 2 water flasks, there are no reactions. The number of ways to choose 2 flasks from the 2 water flasks is:
[
\text{Unfavorable ways} = \binom{2}{2} = 1
]
So, there is only 1 unfavorable outcome (both selected flasks contain water, and thus no reaction occurs).
Step 3: Calculate the Favorable Outcomes
The favorable outcomes are those where at least one flask contains hydrogen peroxide. Since the total number of ways to select 2 flasks is 15, and there is only 1 unfavorable outcome, the number of favorable outcomes is:
[
\text{Favorable ways} = \text{Total ways} – \text{Unfavorable ways} = 15 – 1 = 14
]
Step 4: Calculate the Probability
The probability of a favorable outcome (at least one reaction occurring) is the ratio of favorable outcomes to the total number of outcomes:
[
\text{Probability} = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{14}{15}
]
Final Answer:
The probability that at least one reaction will occur is (\frac{14}{15}), or approximately 0.9333 or 93.33%.
Explanation:
We first calculate the total number of ways to choose 2 flasks from 6, which is 15. Then, we determine the unfavorable outcome where neither of the selected flasks contains hydrogen peroxide, which happens in only 1 case. Subtracting this from the total, we find that 14 of the 15 ways result in at least one reaction. Hence, the probability of having at least one reaction is (\frac{14}{15}).