A man went to Monte Carlo to try and make his fortune. Whilst he
was there he had an opportunity to bet on the outcome of rolling
dice. He was offered the same odds for each of the following
outcomes: At least 1 six with 6 dice. At least 2 sixes with 12
dice. At least 3 sixes with 18 dice.
Two bags contain different numbers of red and blue balls. A ball is
removed from one of the bags. The ball is blue. What is the
probability that it was removed from bag A?
A gambler bets half the money in his pocket on the toss of a coin,
winning an equal amount for a head and losing his money if the
result is a tail. After 2n plays he has won exactly n times. Has he
more money than he started with?
Coin Tossing Games
Set by Dr Susan Pitts, University of Cambridge Statistics
for the Summer 1997 NRICH Maths Club Video-conference.
You and I play a game involving successive throws of a fair
coin. Let H and T denote heads and tails respectively.
I pick HH. Suppose that you pick TH. The coin is thrown
repeatedly until we see either two heads in a row or a tail
followed by a head. In the first case I win; in the second case you
win. What is the probability that you win?
What is the probability that you win if you choose HT? Or TT?
What is the best choice you can make?
What should you choose if I choose TT?
What happens if I choose HT?
Assuming that you always make a choice that maximises your
chance of winning, what should I choose to maximise the probability
that I win?
Now suppose that we look at triples instead of pairs. What is
the probability that you win if I choose HHH and you choose
I have eight possible choices. For each one, can you find a
triple that gives you a better than even chance of winning (i.e. a
triple that makes your probability of winning more than 1/2)?