Question

We know that 10% people who purchase cars buy manual transmissions and 90% people purchase automatics.Three people purchase a car. find the probability distribution for the number who purchase automatics

Answer 1

Obviously the number N of purchased automatics may be any value of 0, 1, 2, 3.

We have a Bernoulli trial - exactly two possible outcomes, "success" (an automatics is purchased) and "failure" (a manual transmission is purchased) and the probability of success is the same every time the experiment is conducted (a car is purchased).

The probability of each result:

$P(N=k)=\begin{pmatrix} n \\ k \end{pmatrix}\cdot p^k \cdot q^{n-k}=\\=\begin{pmatrix} 3 \\ k \end{pmatrix}\cdot 0.1 ^k \cdot0. 9^{3-k}=\\$

$=\cfrac{3!}{k!\cdot(3-k)!}\cdot0.1^k\cdot0. 9^{3-k};$

$P(N=0)=\cfrac{3! }{0!\cdot 3!}\cdot0.1^0\cdot0.9^3=\\=0.9^3=0.729;$

$P(N=1)=\cfrac{3! }{1!\cdot 2!}\cdot0.1^1\cdot0.9^2=\\=3\cdot0.1\cdot0.9^2=0.243;$

$P(N=2)=\cfrac{3! }{2!\cdot 1!}\cdot0.1^2\cdot0.9^1=\\=3\cdot0.1^2\cdot0.9=0.027;$

$P(N=3)=\cfrac{3! }{3!\cdot 0!}\cdot0.1^3\cdot0.9^0=\\=0.1^3=0.001.$