Linear Algera. Here is problem: thr store where I shop has fresh bread 1 day out of 2, and fresh grapes 1 day out of 3. What are the probabilities for a random day being: neither is fresh, the bread but not the grapes, the grapes but not the bread, both.
Gemini:Here's how to solve this probability problem using a bit of linear algebra thinking:
1. Define Events
- Let event "B" represent "fresh bread"
- Let event "G" represent "fresh grapes"
2. Probabilities
- P(B) = 1/2 (bread is fresh 1 day out of 2)
- P(G) = 1/3 (grapes are fresh 1 day out of 3)
3. Assume Independence
We'll assume that the freshness of bread and grapes are independent events. This means the freshness of one doesn't affect the freshness of the other.
4. Calculate Probabilities
Neither Fresh:
- P(not B) = 1 - P(B) = 1 - 1/2 = 1/2
- P(not G) = 1 - P(G) = 1 - 1/3 = 2/3
- P(neither fresh) = P(not B) * P(not G) = (1/2) * (2/3) = 1/3
Bread but not Grapes:
- P(B and not G) = P(B) * P(not G) = (1/2) * (2/3) = 1/3
Grapes but not Bread:
- P(G and not B) = P(G) * P(not B) = (1/3) * (1/2) = 1/6
Both Fresh:
- P(B and G) = P(B) * P(G) = (1/2) * (1/3) = 1/6
Summary of Probabilities:
- Neither Fresh: 1/3
- Bread but not Grapes: 1/3
- Grapes but not Bread: 1/6
- Both Fresh: 1/6
Linear Algebra Connection
While this problem doesn't directly involve matrices or vectors, the concept of independence and the multiplication of probabilities are fundamental to linear algebra. Think of the probabilities as elements of a vector representing the state of the system, and the multiplication as a linear transformation (scaling and combining elements).
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