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Students will make predictions about data by using the outcomes of previously solved probability events.
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Students will understand the probability of an event is from 0-100% and apply this to real-world situations to make informed decisions.
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Students will communicate and justify their inferences when using real-world situations.
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While applying probability to real-world experiments, students will have a basic understanding of theoretical probability and experimental probability.
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Students will understand the relationship of independent and dependent data.
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Students will understand that randomness and fairness of choice are a part of the probability concepts.
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Students will define symmetrical outcomes.
Example:
A traveler approaches a river spanned by bridges that connect its shores and islands. There has been a great storm the night before, and each bridges was as likely as not to be washed away. How probable is it that the traveler can still cross?
In the absence of any additional (bridge structural and wear-and-tear) information, the traveler believes (and we will too) that all the bridges have equal probabilities and independent chances of washing away.
Solution:
The configuration of bridges has a remarkable symmetric structure.
Imagine that there is a boat floating down the stream. Any bridge that withstood the storm will block the passage of the boat. The boat will pass through if a sufficient number of bridges have been washed away. What is the probability that the boat will get through?
The key observation that brings a solution to the problem is that it has the same abstract structure for the traveler and the boat:
For the traveler, bridges 1 and 2 are the entries, whereas bridges 4 and 5 are exits. For the boat the entries are at bridges 1 and 4, the exits are at bridges 2 and 5.
A bridge that has withstood the storm is good for the traveler, but bad for the boat, and vice versa. But each individual event has the probability of 50%. Having an identical abstract structure suggests the following observation. (Writing P(X) for the probability of event X):
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P(traveler crosses)=P(Boat gets through).
But, beyond that, as we just observed, what is good for one is bad for the other, such that
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P(boat gets through)=P(traveler does not cross).
As a consequence of 1 and 3, we get
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P(traveler crosses)=P(traveler does not cross),
with the inevitable conclusion that both probabilities evaluate to 50%.
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