# Quadratic equation and probability: bag contains n yellow balls, 7 white balls

#### bumblebee123

##### New member
PLease can you tell me where I am going wrong. I have a question from the quadratic equation which uses probability. The information I have is as follows:

A bag contains (n+7) balls. n are the number of yellow balls.The other 7 balls are white.

I already have the expression, in terms of n, for the probability of taking a white ball:
7/n+7

I am now told.

There are two people. After John has put the ball back into the ball bag, Mary takes a ball at random from the bag. Given the probability that John and Mary will take balls of different colours is 4/9 prove that 2n^2 - 35n + 98 = 0

So far I have done this:

If they both choose a ball each and the ball are different colours then I need to use the formula for a ball being white and another being yellow and as they are independent probabilities I will need to multiply them, I have done this:

n/n+7 x 7/n+7 = 4/9

I try to get the equation to resemble the 2n^2 - 35n + 98 = 0 and proceed to :

7n/(n+7)^2= 4/9

7n = 4/9 (n+7)^2

63n = 4 (n+7)^2
63n = 4 (n^2 + 14n +49)
63n = 4n^2 + 56n + 196
0 = 4n^2 -7n +196

I see that if I divide my 2 as a common factor I get:

0 = 2x^2 -3.5n - 98

Which is nearly what I need but the coefficient for n is not right.

Please can you help? I don't know where I have gone wrong or if this is a misprint in the book - maybe.

Thank you.

#### Dr.Peterson

##### Elite Member
PLease can you tell me where I am going wrong. I have a question from the quadratic equation which uses probability. The information I have is as follows:

A bag contains (n+7) balls. n are the number of yellow balls.The other 7 balls are white.

I already have the expression, in terms of n, for the probability of taking a white ball:
7/n+7

I am now told.

There are two people. After John has put the ball back into the ball bag, Mary takes a ball at random from the bag. Given the probability that John and Mary will take balls of different colours is 4/9 prove that 2n^2 - 35n + 98 = 0

So far I have done this:

If they both choose a ball each and the ball are different colours then I need to use the formula for a ball being white and another being yellow and as they are independent probabilities I will need to multiply them, I have done this:

n/n+7 x 7/n+7 = 4/9

I try to get the equation to resemble the 2n^2 - 35n + 98 = 0 and proceed to :

7n/(n+7)^2= 4/9

7n = 4/9 (n+7)^2

63n = 4 (n+7)^2
63n = 4 (n^2 + 14n +49)
63n = 4n^2 + 56n + 196
0 = 4n^2 -7n +196

I see that if I divide my 2 as a common factor I get:

0 = 2x^2 -3.5n - 98

Which is nearly what I need but the coefficient for n is not right.

Please can you help? I don't know where I have gone wrong or if this is a misprint in the book - maybe.

Thank you.
If the two balls are different colors, they maybe either WY or YW. You assumed only the first.

You have to double the probability expression you wrote, to take both orders into account. Then you will get the right equation.

#### bumblebee123

##### New member
If the two balls are different colors, they maybe either WY or YW. You assumed only the first.

You have to double the probability expression you wrote, to take both orders into account. Then you will get the right equation.

Thank you so much for your help. I managed to get the answer, thank you.

#### Dr.Peterson

##### Elite Member

Thank you so much for your help. I managed to get the answer, thank you.
Great. This was an easy thing to overlook.

#### Jomo

##### Elite Member

Thank you so much for your help. I managed to get the answer, thank you.

I am glad that you finally solved the problem. You did good work. I do feel that you are not finished, that is, there should have been another part to this problem. How many yellow balls are there?