Question

A basket contains 17 ​eggs, 3 of which are cracked. If we randomly select 10 of the eggs for hard​ boiling, what is the probability of the following​ events?

a. All of the cracked eggs are selected.

b. None of the cracked eggs are selected.

c. Two of the cracked eggs are selected.

Answer 1

$\\\mathbf{a.}\;P(All\; cracked \;eggs\; are \;selected)=\frac{_{3}^{3}\textrm{C}*_{7}^{14}\textrm{C}}{_{10}^{17}\textrm{C}}=\frac{1*3432}{19488}$

$P(All\; cracked \;eggs\; are \;selected)={\color{Red} 0.1761}$

$\mathbf{b.}\;P(None \;of\; the\; cracked \;eggs\; are\; selected)=\frac{_{10}^{14}\textrm{C}}{_{10}^{17}\textrm{C}}=\frac{1001}{19488}$

$P(None \;of\; the\; cracked \;eggs\; are\; selected)={\color{Red} 0.0514}$

$\\\mathbf{c.}\;P(Two \;of\; the \;cracked\; eggs \;are\; selected)=\frac{_{2}^{3}\textrm{C}*_{8}^{14}\textrm{C}}{_{10}^{17}\textrm{C}}=\frac{3*3003}{19488}$

$P(All\; cracked \;eggs\; are \;selected)={\color{Red} 0.4623}$