Problem 1

Let us say that the total human population of earth is 10 million. Out of this 1,000 persons are infected with AIDS and they will die sooner or later. That is 0.01% of earth's population (men and women) has AIDS. Also, assume that earth's human population is growing at the rate of 3% per annum. Now assuming that this deadly AIDS disease is doubling every year it is very likely that:

1. in 10 years time the entire population of earth will be wiped out;
2. in 15 years time the entire population of earth will be wiped out;
3. in 20 years time the entire population of earth will be wiped out;
4. Eventually the AIDS epidemic curve taken together for the entire population of the earth will flatten out and the earth's population will not be destroyed entirely;

( Hint : this is a problem in exponentials. AIDS epidemic in the above problem is increasing exponentially and all exponential curves eventually flatten out over time.)

Problem 2

A sequence of rows, in the shape of a triangle, is given as follows:

Fill in the ? space with the correct numbers

Problem 3

A circle is centred at the origin of a plane where the horizontal axis is and the vertical axis is . At origin the value of the coordinates is . A point P, whose coordinates are , lies outside the circle in the plane . If is the radius of the circle then the distance of the point P from any point on the outermost boundary of the circle will be:

1. 
2. 
3. 
4. None of the above

( Hint : The distance of the point from the centre of the circle is and the radius of the circle is the distance from the centre of the circle to any point on the outermost boundary of the circle.)

Problem 4

How many people should be present in a room before it is more likely than not that two of them will share the same birthday?

1. 15 or more
2. 23 or more
3. 67 or more
4. 145 or more

( Hint : This is the classic converse probability problem. When the second person enters the room the probability that he or she will not share the birthday with the first is 364/365, or 99.7%. When the third person enters the room the probability that he does not share the same birthday with the first two are (364/365)*(363/365) = 99.2%. Work this way until you reach the probability figure of 50%. At that point the problem is solved.)

Problem 5

If is the space and is the probability measure, then the measure of the whole space satisfies:

1. 
2. 
3. 
4. The measure of the whole space is indeterminate;

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