Why most of the minority carriers make it across the base as the base width is small? I am studying about BJT from an online lecture note. I made a question in the image below. The reference in the image is from the two lectures, lecture 18 and lecture 19. 
Could anyone help me with the question in the image?

 A: When the hole concentration at the $x=0$ boundary is held constant at $n_0$ and holes are allowed to diffuse in the $0 < x < W$ region, they each travel a certain distance before being annihilated through recombination. The quantity $L_B$ in your notation represents the hole diffusion length, or the average distance traveled by diffusing holes during their average lifetime $\tau$. 
When the diffusion length is large compared to the base width, $L_B >> W$, it means that most holes travel distances larger than $W$ before recombination, or else, that very few holes recombine while traveling a distance $W$. So the average concentration of holes surviving at the distal boundary of the base is comparable to their original concentration at the opposite boundary. And the lower the width $W$, the less the fraction of holes annihilated within the base. 
If you prefer a quantitive treatment, the hole concentration at time $t$ and distance $x$ from the boundary  is given by Ficks's diffusion laws as 
$$
n(x, t) = n_0 \,\text{erfc}\Big(\frac{x}{L_B}\sqrt{\frac{\tau}{t}}\Big)
$$
where erfc(x) is known as the complementary error function. Do not worry about its exact expression. What is important is that for distances $x$ much smaller than the diffussion length $L_B$, $x/L_B << 1$, its Taylor expansion is just $\text{erfc}(x) = 1 - 2 x/\sqrt{\pi}$. Which means that for small base widths, such that $x/L_B < W/ L_B << 1$, and for time scales on the order of the hole lifetime $t \sim \tau$, the hole concentration varies linearly with the distance $x$, 
$$
n(x) \approx n_0 \Big(1 - 2 \frac{x}{L_B}\Big)
$$
So at the other boundary of the base, for $x = W$, the concentration reads $n(W) \approx n_0 \Big(1 - 2 \frac{W}{L_B}\Big)$, which shows that the lower the width, the higher the surviving concentration. 
