Question

Let \(X\) be \(b(2, p)\) and let \(Y\) be \(b(4, p)\). If \(P(X \geq 1)=\frac{5}{9}\), find \(P(Y \geq 1)\).

Short answer

The probability \(P(Y \geq 1)\) is equal to \( 1 - \left(\frac{2}{3}\right)^4 = \frac{80}{81}\)

Step-by-step solution

Find the value of \(p\)

Given \(P(X \geq 1) = 1 - P(X = 0) = \frac{5}{9}\), we can write the binomial probability for \(X\)=0 as \(P(X=0) = \binom{2}{0}*(p^0)*(1-p)^{2} = (1-p)^{2}\). Therefore we can solve the following equation to find the value of \(p\): \(\frac{5}{9} = 1 - (1 - p)^2\)

Solve for \(p\)

From step 1, we have the equation \(\frac{5}{9} = 1 - (1 - p)^2\). It simplifies to \(1-\frac{5}{9} = (1 - p)^2 \Rightarrow \frac{4}{9} = (1 - p)^2\). Taking the square root of both sides, we find \(p = 1 - \frac{2}{3}\), so \(p = \frac{1}{3}\).

Find the probability \(P(Y \geq 1)\)

We want to find \(P(Y \geq 1)\), we are realizing that it actually is \(1 - P(Y = 0)\). Now that we have \(p\) from step 2, we can solve it: \(P(Y \geq 1) = 1 - P(Y = 0) = 1 - \binom{4}{0}*(p^{0})*(1-p)^{4} = 1 - (1-p)^4 \). Substituting the value of \(p\), we have \(P(Y \geq 1) = 1 - (1-\frac{1}{3})^4 = 1 - \left(\frac{2}{3}\right)^4\)