Question

2. We take a rod of length 1 metre and randomly divide it into two pieces. Write down the sample space for this experiment. Can you plot it? Write down the event E that at least one of the pieces has length strictly bigger than 1/2 metre. 3. How many ways are there to color n different balls with 4 colors?

Answer 1

Let the random variable representing the lengths of the broken stick be $X_1,X_2$. Here $X_1+X_2=1,X_1>0,X_2>0$

The sample space for this experiment is ${\color{Blue} S=\left \{ \left ( X_1,X_2 \right )|X_1+X_2=1,X_1>0,X_2>0 \right \}}$

The sample space s plotted below. It is a straight line.
Since $X_1+X_2=1,X_1>0,X_2>0$ we can say that, one piece is larger than the other and larger piece is larger than or equal to 0.5. The event is ${\color{Blue} E=X_1\geqslant 1/2 \textup{ or }X_2\geqslant 1/2}$ .