Understanding the Problem
We have a random variable X that follows a Binomial distribution, denoted as B(6, 1/3). This means:
- The number of trials (n) is 6.
- The probability of success in each trial (p) is 1/3.
The problem states that K * P(X=4) = P(X=2). We need to find the value of K.
Calculating P(X=4)
Using the Binomial probability formula:
- P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
For P(X=4):
- n choose 4 = 6! / (4!(6-4)!) = 15
- p^4 = (1/3)^4 = 1/81
- (1-p)^(n-k) = (2/3)^2 = 4/9
Thus,
P(X=4) = 15 * (1/81) * (4/9) = 60/729
Calculating P(X=2)
For P(X=2):
- n choose 2 = 6! / (2!(6-2)!) = 15
- p^2 = (1/3)^2 = 1/9
- (1-p)^(n-k) = (2/3)^4 = 16/81
Thus,
P(X=2) = 15 * (1/9) * (16/81) = 240/729
Setting Up the Equation
According to the problem statement:
K * P(X=4) = P(X=2)
Substituting our computed values:
K * (60/729) = 240/729
Solving for K
To isolate K, divide both sides by (60/729):
K = (240/729) / (60/729) = 240/60 = 4
Final Answer
The value of K is 4.