Advanced Probability

Aman verma wrote:Q: Each coefficient in the equation ax^2 + bx + c = 0 is determined by throwing ordinary six faced die . Find the probability that the equation will have real roots

a) 34/161

b)43/216

c)25/161

d) 47/216

e)31/216

An equation ax^2 + bx + c = 0 has real roots when b^2 is greater than 4*a*c

There are 6*6*6 ways to choose coefficients

when b=1 there is no roots no matter what
when b=2 there are roots only when a=1 and c=1 - 1 way
when b =3 (1,1), (2,1) and (1,2) satisfies condition - 3 ways
when b= 4 (1,1), (1,2) (2,1) (2,2) (1,3) (3,1) (1,4) (4,1) satisfies the conditon - 8 ways
when b=5 (1,1),(1,2), (2,1), (1,3), (3,1), (1,4), (4,1), (1,5),(5,1) (1,6) (6,1), (2,3), (3,2), (2,2) - 14 ways
when b=6 (1,1),(1,2), (2,1), (1,3), (3,1), (1,4), (4,1), (1,5),(5,1) (1,6) (6,1), (2,3), (3,2), (2,2) (2,4), (4,2),(3,3) - 17 ways

total of 43 ways

Probability = 43/216

onedayi'll wrote:Can some one please through light on this, i didn;t get this...

Okay I will try to explain

The roots of the equation ax^2 + bx + c = 0 are (-b +/- sqrt(b^2-4ac)/2a

the roots are real when sqrt(b^2-4ac) is real;

that is when b^2-4ac is positive or zero.

Now in this case a b and c are the numbers you get when you roll the dice

You can have 6*6*6 distinct combination of a, b and c and out of which we have to check how many satisfies the condition b^2 => 4ac

So we start with all the possible values of b and represent the pair (a,c) as an ordered pair
When b=1; b^2 =1 b^2 -4ac is always less than zero because the least values a and c can take is 1
when b =2 b^2 = 4 and when a and c are 1 the satisfies the condition that b^2 -4ac =0 that is 1 way (a=1 b=2 c=1)
then we check for b=3 and all the pairs are listed in the above post which can satisfy this condition

onedayi'll wrote:Can some one please through light on this, i didn;t get this...

Not sure where this question is from, but unless you're scoring 900+ I really wouldn't worry about it too much.

Stuart Kovinsky wrote:

onedayi'll wrote:Can some one please through light on this, i didn;t get this...

Not sure where this question is from, but unless you're scoring 900+ I really wouldn't worry about it too much.

yeah it would really take a lot of time and we can't solve this under so limited time..don't worry this would not gngn to come on the D day for sure...