Binomial Theorem

Given the binomial distribution X∼B(6,p) and the condition ‌

P(X=4)

P(X=2)

=‌

1

9

, let's solve for p.
Start with the equation:
9P(X=4)=P(X=2)
For a binomial distribution:
P(X=k)=(

n

k

)pk(1−p)n−k
Using this formula, we set up the probabilities:
‌P(X=4)=(

6

4

)p4(1−p)2
‌P(X=2)=(

6

2

)p2(1−p)4
We substitute these into our equation:
9×(

6

4

)p4(1−p)2=(

6

2

)p2(1−p)4
Calculate the binomial coefficients:
‌(

6

4

)=15
‌(

6

2

)=15
Substitute these values:
9×15p4(1−p)2=15p2(1−p)4

Cancel out the common term 15 :
9p4(1−p)2=p2(1−p)4
Divide both sides by p2(1−p)2 (assuming p≠0 and p≠1 ):
9p2=(1−p)2
Take the square root of both sides:
3p=1−p
Solve for p :
‌3p+p=1
‌4p=1
‌p=‌

1

4

Therefore, the value of p is ‌

1

4

.