Answer to Question #169035 in Statistics and Probability for Bikram Chaudhary

Question

If 10% of a large consignments of eggs are bad, what is the probability distribution of the number of bad eggs in a box of half dozen chosen at random?

Answer 1

$p = 0.1 \Rightarrow q = 1 - p = 0.9$

Using the Bernoulli formula, we find the probabilities that there will be 0, 1, 2, 3, 4, 5 and 6 bad eggs, respectively:

${P_6}\left( 0 \right) = {q^6} = {0.9^6} = {\rm{0}}{\rm{.531441}}$

${P_6}\left( 1 \right) = C_6^1p{q^5} = 6 \cdot 0.1 \cdot {0.9^5} = {\rm{0}}{\rm{.354294}}$

${P_6}\left( 2 \right) = C_6^2{p^2}{q^4} = 15 \cdot {0.1^2} \cdot {0.9^4} = {\rm{0}}{\rm{.098415}}$

${P_6}\left( 3 \right) = C_6^3{p^3}{q^3} = 20 \cdot {0.1^3} \cdot {0.9^3} = {\rm{0}}{\rm{.01458}}$

${P_6}\left( 4 \right) = C_6^4{p^4}{q^2} = 15 \cdot {0.1^4} \cdot {0.9^2} = {\rm{0}}{\rm{.001215}}$

${P_6}\left( 5 \right) = C_6^5{p^5}q = 6 \cdot {0.1^5} \cdot 0.9 = {\rm{0}}{\rm{.000054}}$

${P_6}\left( 6 \right) = {p^6} = {0.1^6} = {\rm{0}}{\rm{,000001}}$

We have the probability distribution:

$\begin{matrix} X&0&1&2&3&4&5&6\\ p&{{\rm{0}}{\rm{.531441}}}&{{\rm{0}}{\rm{.354294}}}&{{\rm{0}}{\rm{.098415}}}&{{\rm{0}}{\rm{.01458}}}&{{\rm{0}}{\rm{.001215}}}&{{\rm{0}}{\rm{.000054}}}&{{\rm{0}}{\rm{,000001}}} \end{matrix}$