Interesting and disturbing trivia: In most countries the ratio of boys to girls is about \displaystyle{1.04}:{1}1.04:1, but in China it is \displaystyle{1.15}:{1}1.15:1.]

Interesting and disturbing trivia: In most countries the ratio of boys to girls is about $\displaystyle{1.04}:{1}$, but in China it is $\displaystyle{1.15}:{1}$

$1.15:1$.] 

Solution

The probability of getting a boy is 

$\displaystyle\frac{1.09}{{{1.09}+{1.00}}}={0.5215}$

Let $\displaystyle{X}=$ number of boys in the family.

Here,

$\displaystyle{n}={6}$,
$\displaystyle{p}={0.5215}$,
$\displaystyle{q}={1}-{0.52153}={0.4785}$

When $\displaystyle{x}={3}$:

$\displaystyle{P}{\left({X}\right)}$ $\displaystyle={{C}_{{x}}^{{n}}}{p}^{x}{q}^{{{n}-{x}}}$ $\displaystyle={{C}_{{3}}^{{6}}}{\left({0.5215}\right)}^{3}{\left({0.4785}\right)}^{3}$ $\displaystyle={0.31077}$

When $\displaystyle{x}={4}$:

$\displaystyle{P}{\left({X}\right)}$ $\displaystyle={{C}_{{4}}^{{6}}}{\left({0.5215}\right)}^{4}{\left({0.4785}\right)}^{2}$ $\displaystyle={0.25402}$

When $\displaystyle{x}={5}$:

$\displaystyle{P}{\left({X}\right)}$ $\displaystyle={{C}_{{5}}^{{6}}}{\left({0.5215}\right)}^{5}{\left({0.4785}\right)}^{1}$ $\displaystyle={0.11074}$

When $\displaystyle{x}={6}$:

$\displaystyle{P}{\left({X}\right)}$ $\displaystyle={{C}_{{6}}^{{6}}}{\left({0.5215}\right)}^{6}{\left({0.4785}\right)}^{0}$ $\displaystyle={2.0115}\times{10}^{ -{{2}}}$

So the probability of getting at least 3 boys is:

$\displaystyle\text{Probability}={P}{\left({X}\ge{3}\right)}$

$\displaystyle={0.31077}+{0.25402}+$ $\displaystyle{0.11074}+$ $\displaystyle{2.0115}\times{10}^{ -{{2}}}$

$\displaystyle={0.69565}$

NOTE: We could have calculated it like this:

$\displaystyle{P}{\left({X}\ge{3}\right)}$ $\displaystyle={1}-{\left({P}{\left({x}_{{0}}\right)}+{P}{\left({x}_{{1}}\right)}+{P}{\left({x}_{{2}}\right)}\right)}$