Question

Each of the $n$ passengers sitting in a bus may get down from it at the next stop with probability $p$. Moreover, at the next stop either no passenger or exactly one passenger boards the bus. The probability of no passenger boarding the bus at the next stop being $p_o$. Find the probability that when the bus continues on its way after the stop, there will again be n passengers in the bus

• A. $(1-p{)}^{n-1}.[p_o(1-p)+np(1-p_0)]$
• B. $1-(1-p{)}^{n-1}.[p_o(1-p)+np(1-p_0)]$
• C. $p^n.\left[p_o\right(1-p)+np(1-p_0\left)\right]$
• D. $1-p^n.\left[p_o\right(1-p)+np(1-p_0\left)\right]$

Answer 1

The correct option is A $(1-p{)}^{n-1}.[p_o(1-p)+np(1-p_0)]$

N passenger = 2 cases possible

(I) No passenger gets down at next stop + no passenger boards the bus

(II) only One passenger gets down at next stop + 1 passenger boards the bus.

$P\left(n\right)=P\left(I\right)+P\left(II\right)$

$P\left(n\right)=(1-p{)}^n{p}_o{+}^n{C}_{1}p(1-p{)}^{n-1}(1-p_o)$

$=(1-p{)}^{n-1}.[p_o(1-p)+np(1-p_0)]$