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 find the probability that at least one reaction occurs when yeast is added to two of the flasks, we first need to identify the total number of flasks and the specific flasks containing hydrogen peroxide (H₂O₂).
Total flasks: 6
Flasks with water: 2
Flasks with hydrogen peroxide: 4
When yeast is added to two flasks, the probability of a reaction occurring depends solely on whether yeast is added to the flasks containing hydrogen peroxide. To solve this, we can use complementary counting—first determining the probability that no reactions occur and then subtracting that from 1.
Step 1: Calculate total ways to choose 2 flasks from 6.
The number of ways to choose 2 flasks from 6 is calculated using the combination formula:
[
\text{Total combinations} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
]
Step 2: Calculate ways to choose 2 flasks that do not cause a reaction.
The only flasks that do not cause a reaction are the 2 flasks containing water. The number of ways to choose 2 flasks from the 2 water flasks is:
[
\text{Water combinations} = \binom{2}{2} = 1
]
Step 3: Calculate the probability of no reaction.
The probability of choosing 2 water flasks (no reaction) out of the total ways to choose 2 flasks is:
[
P(\text{no reaction}) = \frac{\text{Water combinations}}{\text{Total combinations}} = \frac{1}{15}
]
Step 4: Calculate the probability of at least one reaction.
Finally, to find the probability of at least one reaction occurring, we subtract the probability of no reaction from 1:
[
P(\text{at least one reaction}) = 1 – P(\text{no reaction}) = 1 – \frac{1}{15} = \frac{14}{15}
]
Conclusion:
Thus, the probability that at least one reaction occurs when yeast is added to the flasks is (\frac{14}{15}), or approximately 93.33%. This high probability reflects that with four flasks containing hydrogen peroxide, adding yeast to two randomly chosen flasks makes it highly likely that at least one of those will cause a reaction.