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Help plz!

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Randomly choose two numbers between 0 and 1. What is the probability that their mean is no greater than 2/3?

Aug 6, 2020

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Randomly choose two numbers between 0 and 1.
What is the probability that their mean is no greater than 2/3?

$$\text{Let number one = x } \\ \text{Let number two = y } \\ \text{Let the mean = \dfrac{x+y}{2} }$$

Their mean is no greater than $$\dfrac{2}{3}$$:
$$\dfrac{x+y}{2} \le \dfrac{2}{3}$$

$$\begin{array}{|rcll|} \hline \dfrac{x+y}{2} &=& \dfrac{2}{3} \\ x+y &=& \dfrac{4}{3} \quad | \quad y=1 \\ x+1 &=& \dfrac{4}{3} \\ x &=& \dfrac{4}{3}-1 \\ \mathbf{x} &=& \mathbf{\dfrac{1}{3}} \\ \hline \end{array} \begin{array}{|rcll|} \hline \dfrac{x+y}{2} &=& \dfrac{2}{3} \\ x+y &=& \dfrac{4}{3} \quad | \quad x=1 \\ 1+y &=& \dfrac{4}{3} \\ y &=& \dfrac{4}{3}-1 \\ \mathbf{y} &=& \mathbf{\dfrac{1}{3}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \text{The probability} &=& \dfrac{A_{\color{red}red}}{A} \\\\ && A_{\color{red}red} = \dfrac{\left(1+\frac13\right)}{2}*\dfrac23+\dfrac13*1 \\\\ && \mathbf{A_{\color{red}red} = \dfrac{7}{9}} \\\\ && A = 1*1 \\\\ && \mathbf{A = 1} \\\\ \text{The probability} &=& \dfrac{\frac{7}{9}}{1} \\\\ \mathbf{\text{The probability}} &=& \mathbf{\dfrac{7}{9}} \\ \hline \end{array}$$

Aug 10, 2020