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In this lesson
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Conditioning and independence
A probability depends on the information available. Conditioning means restricting attention to outcomes consistent with what you learned, then renormalizing their probabilities.
In this model, both dice are fair and independent. Ask whether the sum is at least 10. Without additional information, 6 of 36 ordered outcomes qualify. Now learn that the first die is at least 4. Only 18 outcomes remain, and the same six favorable pairs are among them. The conditional probability doubles to 6/18 = 1/3.
10 / 36 = 27.78%. Both dice are independent and fair. Conditioning removes incompatible outcomes before you count.
The denominator is the lesson
Set the first-die threshold to 4 and the sum threshold to 10. The marked “excluded” cells are no longer possible given the information. They still existed in the original sample space; conditioning has not changed how the dice were rolled.
In symbols, P(A | B) = P(A and B) / P(B), provided P(B) is positive. Here A is “sum at least 10” and B is “first die at least 4”. The joint probability is 6/36 and P(B) is 18/36, giving 1/3.
Make a prediction
Given that the first die is 6, what is the chance the sum is at least 10?
Explore the answer
The second die must be 4, 5, or 6. Three of its six outcomes qualify, so the answer is 1/2. Set the first-die threshold to 6 to verify it.
Independence is a claim to check
Events A and B are independent when P(A and B) = P(A)P(B). If P(B) > 0, this is equivalent to P(A | B) = P(A).
The two individual dice are independent by assumption. But their sum and the first die are dependent: a large first die makes a large sum more likely. Derived quantities can be dependent even when the original inputs are independent.
Independence is also different from mutually exclusive events. Two positive-probability events that cannot occur together cannot be independent: observing either rules the other out.
Reverse the question carefully
P(A | B) and P(B | A) usually differ. Among all outcomes with a sum at least 10, the first die is always at least 4. Thus P(B | A) = 1, although P(A | B) = 1/3.
Bayes’ theorem relates these two directions using their base rates. The base-rate paradox shows why a highly accurate test can still yield many false alarms. Monty Hall adds another ingredient: the information-revealing process itself must be part of the model.
Further reading
ProbabilityCourse: formal definitions and worked examples.