Sup pose that 6\% of the people with blood group O are left handed and 10\% of those with other blood groups are left handed, 30\% of the people have blood group 0 . If a left handed person is selected at random, what is the probability that he/she will have blood group O?

Blood group 'O’ Other than blood group 'O'

I. Number of people 30\% 70\%

II Percentage of left handed people 6\% 10\%

${E_1} =$ Event that the person selected is of blood group ${\rm{O}}$

${E_2} =$ Event that the person selected is of other than blood group O

$\left( {{E_3}} \right) =$

Event that selected person is left handed
$\therefore$ $P\left( {{E_1}} \right) = 0.30,$ $P\left( {{E_2}} \right) = 0.70$

$P\left( {{E_3}/{E_1}} \right) = 0.06$ and $P\left( {{E_3}/{E_2}} \right) = 0.10$

By using Baye's theorem,
$P\left( {{E_1}/{E_3}} \right) = \frac{{P\left( {{E_1}} \right) \cdot P\left( {{E_3}/{E_1}} \right)}}{{P\left( {{E_1}} \right) \cdot P\left( {{E_3}/{E_1}} \right) + P\left( {{E_2}} \right) \cdot P\left( {{E_3}/{E_2}} \right)}}$

$= \frac{{0.30 \times 0.06}}{{0.30 \cdot 0.06 + 0.70 \cdot 0.10}}$

$= \frac{{0.0180}}{{0.0180 + 0.0700}}$
$= \frac{{0.0180}}{{0.0880}} = \frac{{180}}{{880}} = \frac{9}{{44}}$