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=‌