Conditional probability and independence
Restrict the sample space when new information is given.
Before you begin
- Fractions and probability
- Two-way tables or set notation
Recalculate within the given group
P(A given B) asks for the proportion satisfying A among outcomes already known to satisfy B. Divide P(A and B) by P(B), provided P(B) is nonzero. In a table of people, count the overlap for the numerator and the entire B group for the denominator. Reversing the condition usually changes that denominator.
Test independence
Independent events satisfy P(A given B) = P(A), when the conditional probability is defined. Equivalently, P(A and B) = P(A)P(B). Mutually exclusive events have no overlap; if both have positive probability, they are not independent. Learning that one occurred then makes the other impossible.
Worked example
Of 40 students, 18 study art and 10 study both art and music. Given that a student studies art, find the probability they study music.
Show the worked solution
- The condition restricts the sample space to the 18 art students.
- Ten of these students also study music.
- Divide 10 by 18 and simplify.
5/9.
Try it yourself
If P(A) = 0.4, P(B) = 0.5 and A and B are independent, find P(A and B).
Common mistakes
- P(A given B) and P(B given A) are not interchangeable.
What can you explain now?
A fair die is rolled. Given that the result is even, find the probability it is greater than 3.
Compare with the explanation
2/3.
The restricted equally likely outcomes are 2, 4 and 6. Two of these, 4 and 6, are greater than 3.
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Common Core mathematics standards ↗