Conditional Probability

Conditional Probability, probability, tree diagram

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`Conditional Probability
Recap: Independent events.
If events A and B are independent, then the probability of B happening does not depend upon whether A
has happened or not. Therefore
and
.
A tree diagram can be drawn:
Dependent events:
If A and B are dependent events, the probability of B happening will depend upon whether A has
happened or not.
We therefore have to introduce conditional probabilities in the tree diagram (as shown below):

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The rule for combining probabilities for dependent events is
P(A and B) = P(A) × P(B | A)
This is equivalent to saying
Example:
Every morning I buy either The Times or The Mail. The probability that I buy The Times is ¾ and the
probability that I buy The Mail is ¼. If I buy The Times, the probability that I complete the crossword is
, whereas if I buy The Mail the probability that I complete the crossword is .…read more

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Notation: T = buy The Times M = buy The Mail C = complete crossword
a) From the tree diagram,
P(complete crossword) = P(T and C) + P(M and C) =
b) The probability that I bought The Mail given that I completed the crossword is given by
Example 2:
0.1% of the population carry a particular faulty gene.
A test exists for detecting whether an individual is a carrier of the gene.…read more

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Note: This example highlights the difficulty of detecting rare conditions or diseases.
Past Examination Question: (OCR)
Students have to pass a test before they are allowed to work in a laboratory. Students do not retake the
test once they have passed it. For a randomly chosen student, the probability of passing the test at the first
attempt is 1/3. On any subsequent attempt, the probability of failing is half the probability of failing on the
previous attempt.…read more

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