What is your probabiltiy of winning?

Question

Consider a square dart board with a circle inscribed in the square.Assume that you always hit the board and that each point of the square is equally likely to be hit.

a) if you throw a dart, what is your probaility of winning? explain you answer.

 

b)Suppose if your dart hits outside of the circle you win, if outside you lose. You throw 7 darts, what is the probability winning at least 4 times? explain the method, round to the nearest hundredrth.

Answer 1

Angelina, 

 

First consider the ratio of the areas of the circle to the inscribed square...

You are guaranteed to throw within the square so if we set its area=1 then the

area of the inscribed circle can represent the probability, p, of a dart landing there, just

as the area of the square represents the probability = 1 of landing within it...certainty.

 

The sides of the square measure 1, since s^2=area of square=1

The diameter of the circle is therefore 1 and its radius is 0.5...

Its area is pi*r^2=p=pi/4 (just over 3/4)

 

Using the probability formula for Bernouilli trials,

 

P(k successes in n trials)=C(n,k) p^k q^(n-k)

 

we can determine the probability of winning exactly 4 times as follows:

 

P(4,7)=(7!/(4!*3!))*(pi/4)^4*(1-pi/4)^3

 

Since the requirement is to win at least four times, 5, 6 or 7 times also satisfy.

 

You must therefore calculate and sum all four of these probabilities.