Let X be a B(2, p) and Y be an independent B(4, p). If `P(X ge 1)=5//9`, then find `P(Y ge1)`.

Answer

To solve the problem, we need to find \( P(Y \geq 1) \) given that \( P(X \geq 1) = \frac{5}{9} \) where \( X \) follows a binomial distribution \( B(2, p) \) and \( Y \) follows a binomial distribution \( B(4, p) \).

### Step-by-step Solution:

1. **Understand the problem**: We have two independent random variables \( X \) and \( Y \). The variable \( X \) follows a binomial distribution with parameters \( n = 2 \) and \( p \), and \( Y \) follows a binomial distribution with parameters \( n = 4 \) and \( p \).

2. **Use the given information**: We know that \( P(X \geq 1) = \frac{5}{9} \). We can express this probability in terms of \( P(X = 0) \):
\[ P(X \geq 1) = 1 - P(X = 0) \]
Therefore,
\[ P(X = 0) = 1 - P(X \geq 1) = 1 - \frac{5}{9} = \frac{4}{9} \]

3. **Calculate \( P(X = 0) \)**: For a binomial distribution \( B(n, p) \), the probability of getting 0 successes is given by:
\[ P(X = 0) = \binom{n}{0} p^0 (1-p)^n = (1-p)^2 \]
Setting this equal to \( \frac{4}{9} \):
\[ (1 - p)^2 = \frac{4}{9} \]

4. **Solve for \( p \)**: Taking the square root of both sides gives:
\[ 1 - p = \frac{2}{3} \quad \text{or} \quad 1 - p = -\frac{2}{3} \text{ (not valid)} \]
Thus,
\[ 1 - p = \frac{2}{3} \implies p = 1 - \frac{2}{3} = \frac{1}{3} \]

5. **Find \( P(Y \geq 1) \)**: Now we need to calculate \( P(Y \geq 1) \):
\[ P(Y \geq 1) = 1 - P(Y = 0) \]
For \( Y \) which follows \( B(4, p) \):
\[ P(Y = 0) = (1 - p)^4 \]
Substituting \( p = \frac{1}{3} \):
\[ P(Y = 0) = \left(1 - \frac{1}{3}\right)^4 = \left(\frac{2}{3}\right)^4 = \frac{16}{81} \]

6. **Calculate \( P(Y \geq 1) \)**:
\[ P(Y \geq 1) = 1 - P(Y = 0) = 1 - \frac{16}{81} = \frac{81 - 16}{81} = \frac{65}{81} \]

### Final Answer:
\[ P(Y \geq 1) = \frac{65}{81} \]