The probability that Ashley drives faster than the speed limit (event A) is 0.34, and the probability that he gets a speeding ticket (event B) is 0.22. The probability that he drives faster than the speed limit, given that he has gotten a speeding ticket, is 1. Are events A and B dependent or independent?

Question

Vocaloid · Answer

for independent events P(A and B) = P(A) * P(A | B) where P(A | B) is the probability of "A given B"

perform the appropriate substitutions to determine if this equation is true or not for the given probabilities

sillybilly123 · Answer

You're given:

$P_A = 0.34$

$P_B = 0.22$

but that doesn't really matter BECAUSE you are also told that: "The probability that he drives faster than the speed limit, **given that** he has gotten a speeding ticket, is **1** .....  well, that should really set the alarm bells ringing.

ie $P_{A|B} = 1$.

So if B happens, A is guaranteed to happen.  In plain-speak, you should be wondering how these can be independent events.

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ASIDE:
more generally, events A and B are **independent** if any **one** of these three applies (they're all the same thing):

$P_{(A \text{ and }  B)} = P_A P_B$
$P_{A|B} = P_A$		
$P_{B|A} = P_B$

That is because, for dependent events, Voca's formula applies:

$P_{(A \text{ and } B)} = P_A  P_{B | A} = P_B  P_{A | B}$

And the terms breakdown.