How can I directly intuit E(Gold Ball Draw) = $1E6(69 - B)/BT?

[scherz0]

So the expectation for this part of the lottery is [math]\frac1T\left(1 + \frac 1B(69-2B)\right)\cdot \$1000000 = \$1000000\cdot\dfrac{69 - B}{BT}$[/math]

 

1. Disregarding the messier LHS, how can I intuit the simpler RHS?

 

I know [math]\color{green}{\dfrac{1E6}{T} = \Pr(}[/math] winning $1E6 in the Gold Ball Draw).  2. But why multiply this [math]\color{green}{\dfrac{1E6}T}[/math] by [math]\color{red}{\dfrac{69 - B}B}[/math]?

 

3. What does [math]\color{red}{\dfrac{69 - B}{B}}[/math] mean?