Worked solution · Exam P
This Exam P sample reference tests Conditional Outcome Counting. There are 15 ordered dice outcomes with the red die larger. Nine of them have opposite parities and hence an odd sum, giving 9/15=3/5, choice E.
Independent solution
Setup
Conditioning on the red result exceeding the green result leaves 15 equally likely ordered outcomes.
Model
An odd sum requires the dice to have opposite parity. Counting the qualifying red-greater ordered pairs gives nine outcomes.
Compute
The conditional probability is therefore nine out of 15, or three fifths.
Answer
The conditional probability is three fifths.
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Original practice · fully worked
Two fair eight-sided dice numbered 1 through 8 are rolled, one red and one green. Given that the red result exceeds the green result, find the probability that their sum is odd.
There are C(8,2)=28 ordered outcomes with red greater. Sixteen have opposite parity, so the conditional probability is 16/28=4/7, choice D.
Setup
There are 28 ordered outcomes in which the red eight-sided die exceeds the green die.
Model
The sum is odd exactly when the two results have opposite parity. Counting those red-greater pairs gives 16 favorable outcomes.
Compute
Dividing 16 by 28 reduces to four sevenths.
Answer
The conditional odd-sum probability is four sevenths.
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