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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.

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Balls and Bags

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?

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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?

Like Father Like Son

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

 

 

Alice and Bob are happily married, and decide to have children. Alice is proud of her long blond hair and brown eyes. Bob is also happy with his brown eyes, which he inherited from his auburn father, although his mother was green - eyed.

 

Alice knows that there is a small chance she might have a daughter looking like her (Alice's) mother, and that her daughter is much more likely to look like her. Bob, on the other hand, believes he won't have a son who reminds him (completely) of his father.

 

1. Can you determine the colour of Bob's hair?

2. Can you describe the eye colour of both Alice's parents?

3. How likely is it that Alice and Bob will have a blond son with green eyes?

4. How likely is it that they will have an auburn daughter?

 

Basic genetics:  In the 19th century, a Czech monk, Gregor Mendel, experimented with pea plants.  He noticed that plants were either tall or short, but when he crossbred a tall plant with a short one he didn't get a medium sized plant - he got a tall one.  However, when he crossbred these new tall plants, sometimes it would result in a short one.  He concluded that 'tall' was dominant and 'short' recessive.

Experiments with true-bred green and yellow pea plants led him to conclude that each parent contributed either a G or a Y characteristic (gene), and that G is dominant over Y, so that GG, GY, and YG plants all appear green, and only YY plants appear yellow.  He concluded that crossing a GG plant with a YY plant would result in offspring which all appeared to be green.  However in the following generation, a GY or YG plant could give its offspring a Y gene rather than a G one.  If the GY or YG plant was cross-bred with a YY plant, not all of the offspring would be green.

For this problem, you may assume that the genes determining a person's hair and a person's eye colour are located in different chromosomes (so are independent of each other) and that the genes for green eyes and blond hair are recessive.
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