This test is a sample test; if I gave such a test, it would be a one-day
take-home.
Five fair coins are tossed. What is the probability of observing exactly
two heads? of observing at least two heads? of not observing any heads?
You roll two fair (6-sided) dice. What is the expected value of the
sum of the two top faces (you have to add/multiply quite a few numbers, so
be patient and carry it through)? Now, if you rolled a pair of identical
values, you are allowed to roll a third fair die and add its top face value to
your sum (for instance, if you rolled 3,3, you can roll a third die; if you
obtain a 4, then your total sum is now 10); what is now the expected sum?
You are dealt a hand of 7 cards from a well shuffled deck of 52 cards.
What is the probability that your hand contains three cards with the same
face value? all four such cards? Assume you have to pay $5 to have a hand
dealt to you, that you lose your money if you have no triples, get $10 (and
thus earn $5) if you have a triple, and get $55 (and thus win $50) if you
have four such cards (in the highly improbable case where you have 4 cards
of one value and 3 cards of another value, the triple is ignored); what
is your expected loss/gain in this game?
Your cat randomly rearranges the letters of your word in Scrabble -- that
is, what you get is a random reordering of the letters of your word.
If you word was "effective", what is the probability that the resulting
string does not contain a consecutive pair of identical letters?