Ratio of boys and girls.

In a country in which people only want boys, every family continues to have children until they have a boy. if they have a girl, they have another child. if they have a boy, they stop. what is the proportion of boys to girls in the country?

Solution:

It doesn't matter how many kids a family has. The odds of having a boy or girl are always the same. So number of boys and girls in the village is same.

Ratio of boys and girls.

In a country in which people only want boys, every family continues to have children until they have a boy. if they have a girl, they have another child. if they have a boy, they stop. what is the proportion of boys to girls in the country?

Solution:

50:50.

probability of a boy/girl is always 2 irrespective of the number of conditions.

for the above problem we can prove it with simple probability theory.

Lets assume P(b) be the probability of having a boy. P(g) be the probability of having a girl.

Total expectation of a boy is equal to (probability of having a boy at first instance + probability of having a girl then a boy + probability of having 2 girls followd by a boy + etc... )
E(b) = 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ......
it comes down to 1 ( infinite series sum = a/1-r = 1/2/1-1/2 = 1).

E(g) = 0 + 1/2 + 1/4 + 1/8....
it comes down to 1.

It means there are equal number of girls and boys in the village.

Number coding

Assume 9 is twice 5; how will you write 6 times 5 in the same system of notation

Solution:

9 is twice 5. So 5 is 4.5.
6 times 5 should be 6*4.5 = 27

Distance between trains

One train runs from A to B at 105 miles per hour; the other runs from B to A at 85 miles per hour. How far apart were the two trains 30 minutes prior to their crossing?

Solution:

Two trains are 95 miles apart 1/2 Hr before they crossed each other

Relative Speed of the trains = 85+105 = 190MPH

One hour before they crossed, they would have been 190 miles apart.

Distance between the trains 1/2 hr before they crossed would be 190/2 = 95miles

Probability of observing a car

If the probability of observing a car in 30 minutes on a highway is 0.95, what is the probability of observing a car in 10 minutes (assuming constant default probability)?

probability of not observing a car in 30 mins = probabality of not observing a car in first 10 mins * not obs in next 10 mins * not obs in last 10 mins

lets assume observing a car in 10 mins = p

probability of not observing the car in 10 mins = 1-p

probability of not observing the car in 30 mins = (1-p)^3

so (1-p)^3 = 0.05

1-p = (0.05)^1/3.
p = 1- (0.05)^1/3.
p = 1- 0.37
p = 0.63.