Question

1.    A square is inscribed in a circle.

a) What is the probability that a point located at random in the interior of the circle turns out to be also interior to the square?

b) What is the probability that of 10 points located at random independently of each other in the interior of the circle, four fall into the square, three on one segment and one each on the remaining three segments?

Answer 1

a)

here probability that a point located at random in the interior of the circle turns out to be also interior to the square

=(area of square)/area of cirlce =2r²/(r²)=2/=0.6366

b)

probability of falling into segments =1-2/=0.3634

hence probability of falling into one of four segments =0.3634/4=0.0908

therefore from multinomial distribution:

probability that of 10 points located at random independently of each other in the interior of the circle, four fall into the square, three on one segment and one each on the remaining three segments

=

=0.009307