### Gambling at Monte Carlo

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.

### Marbles and Bags

Two bags contain different numbers of red and blue marbles. A marble is removed from one of the bags. The marble is blue. What is the probability that it was removed from bag A?

### Coin Tossing Games

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

# Two Girls

##### Stage: 4 Short Challenge Level:

Let the number of boys in the class be $x$.

Hence $\frac{10}{10+x}\times\frac{9}{9+x} = 0.15 = \frac{3}{20}$.

Simplifying gives $1800=3(10+x)(9+x)$ and then $x^{2}+19x-510=0$.

Factorising gives $(x+34)(x-15)=0$ and, since $x\not=-34$, $x=15$.

Alternatively, let the number of students in the class be $x$.

Hence $\frac{10}{x}\times\frac{9}{x-1} = 0.15 = \frac{3}{20}$.

Simplifying gives $600=x(x-1)$ and then $x^{2}-x-600=0$.

Factorising gives $(x+24)(x-25)=0$ and, since $x\not=-24$, $x=25$.

Therefore the number of boys in the class is 15.

Students from Comberton Village College sent us these solutions.

This problem is taken from the UKMT Mathematical Challenges.
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