Player A and B bet on the total roll of two normal dice. Player A bets that
a 12 will be rolled first. Player B bets that two 7s will be rolled consecutively
first. They keep rolling until one person wins? What is the probability
A will win?
(Answer),
(Solution).
Suppose the two players from problem 141 do not wish to stop with one bet
resolved but that they play through millions of rolls of the dice. All
rolls are one continuous stream, in other words they don't start over after
a bet is resolved. If you were to jump ahead x rolls (where x>0) what
is the probability A would win the next bet?
(Answer),
(Solution).
Consider the contest posed in problem 142. What is the probability A wins
a bet on the xth roll (where x>0), what is the probability for B?
(Answer),
(Solution).
The answer to problem 143 seems to imply that A and B have equal long term odds of
winning. However the answer to problem 142 seems to imply that A always has the
advantage. What is the explanation behind this apparant paradox?
(Answer).
If A and B, from problems 141, play through one million rolls how much can A expect to be ahead if they each wager $1 for
each bet resolved. If they play through one million bets resolved how much can A expect to be
ahead?
(Answer),
(Solution).
How can the contest described in 141 end fairly so that neither player has an advantage?
(Answer).
What is the expected number of random numbers, uniformally distributed from 0 to 1,
needed for the sum to be greater than 1?
(Answer),
(Solution).
What is the number of possible combinations in the rubik's cube?
When you get to the island you find the gallows are gone. Without digging at random is
it possible to figure out the location of the treasure chest? If so where is it?
(Answer),
(Solution).
You find an old treasure map to the location of a buried treasure chest on a deserted island. The map states on this island are an oak tree, a palm tree, and a gallows. You are to start at the gallows, walk toward the palm tree while counting your steps. Then turn 90 degrees to the left and walk the same number of steps. At that spot drive a stake. Then walk from the gallows to the oak tree, again measuring your steps. When you get to the oak tree turn 90 degrees to the right and walk the same number of steps, then drive a stake at that spot. Then find the midpoint between the two stakes and dig.
ABCDE * 4 = EDCBA. Solve for A,B,C,D, and E where each is a unique
integer from 0 to 9.
(Answer),
(Solution).
Alice: I am insane.
You are also given that:
Describe the four mathematicians.
(Answer),
(Solution).
Four mathematicians have the following conversation:
Bob: I am pure.
Charlie: I am applied.
Dorothy: I am sane.
Alice: Charlie is pure.
Bob: Dorothy is insane.
Charlie: Bob is applied.
Dorothy: Charlie is sane.
In a backyard lies a block of wood 9" by 9" by 22". The long edge of the board
lies along a north/south direction. An ant is sitting on the south end of the board,
half way up vertically, and 1" from the east edge. A spider is sitting on the north
end of the board, half way up vertically, and 1" from the west edge. The spider can
crawl at a rate of 1" per minute. The ant figures that it will take 1+22+8=31 minutes
for the spider to reach him so he dozes off for a 30 minute nap. Just as the ant
wakes up the spider kills him. By what route did the spider take to get to the ant
in 30 minutes?
(Answer).
A loaded coin has probability of landing on heads of 0.6 and for tail 0.4.
Winning bets on heads pay even money. With an initial fortune of $10 and
betting $1 at a time what is the probability of eventual ruin assuming you
kept playing until being ruined? (Answer), (Solution).
Suppose before taking the first bet of the previous problem you consult your
crystal ball which says you will eventually be ruined. Being a compulsive
gambler you play anyway until ruin. What is the ratio of total heads to
total flips?
(Answer), (Solution).
In the picture above is an equilateral triangle inscribed in a circle. Given that the radius of the circle is 1 what is the length of a and b?
(Answer), (Solution).
Two boats on opposite sides of a river head towards each other at different speeds. When they pass each other the first time they are 700 yards from one shoreline. They continue to the opposite shoreline, turn around, and move towards each other again. When they pass the second time they are 300 yards from the other shoreline. How wide is the river? Assume both boats travel at a constant speed and ignore factors such as turn-around time and the current of the river.
(Answer), (Solution).
Two players have made it to Final Jeopardy. Player A has $7000 and player B has $5000. Each player has a 60% of answering correctly and both know this to be true of their opponent. For the sake of simplicity assume a tie goes to player A. What should be the betting strategy of each player? You may assume that each player has a way to draw a random number if needed. (Answer).
Four men and four women are shipwrecked on a deserted island. Eventually each person falls in love with one person and is loved by one person. You are given the following information:
Who loves who?
(Answer).
Your paint inventory consists of 60 gallons of blue, 40 gallons of red, and 30 gallons of yellow. To make purple paint you mix equals parts of blue and red. To make orange paint you mix equal parts of red and yellow. To make green paint you mix equal parts of blue and yellow. Purple paint sells for $6 a gallon, orange for $20, and green for $9. There is a fixed disposal charge for every unused gallon. How much of each of purple, orange, and green paint should you mix to maximize profits if the disposal cost is (a) $4 per gallon, and (b) $6 per gallon.
(Answer).