Is the contrapositive of "if the probability of an event
is 0.1, then the event is unlikely to occur" true or
false?
The contrapositive of a conditional
statement logically always has the same truth value as the conditional statement. Since
the conditional statement is true (the probability p of an event is
assigned a number `0<=p<=1` with 0 impossible and 1 guaranteed, so p=.1 is
very unlikely), the contrapositive must also be true.
You
would have to be careful in how you worded the contrapositive, as the statement "An
event is likely to occur if the probability of the event is not .1" is false. A better
wording might be an event is likely to occur if the probability of an event is .9. Here
we use the fact that the negation of never is always, and the negation of addition is
subtraction, so we replace 0+.1 with 1-.1.
So
with careful wording the contrapositive is true.
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