Question

A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event "5 appears on the first-die". Find P (E/F). If F is the event "5 appears on at least one die", find P (E/F).

Answer 1

Consider the given events.
E = The sum of the numbers on two dice is 10 or more
F = 5 appears on first die

Clearly,
E = {(4, 6),(5, 5),(5, 6),(6, 4), (6, 5), (6, 6)}
F = {(5, 1), (5, 2), (5, 3), (5, 4) (5, 5), (5, 6)}

$$\begin{gathered} \mathrm{Now}, \\ E\cap F=\left\{\left(5,5\right),\left(5,6\right)\right\} \\ \therefore \mathrm{Required}\mathrm{probability}=P\left(E/F\right)=\frac{n\left(E\cap F\right)}{n\left(F\right)}=\frac{2}{6}=\frac{1}{3} \end{gathered}$$

Second case:
Consider the given events.
E = The sum of the numbers on two dice is 10 or more
F = 5 appears on a die at least once

Clearly,
E = {(4, 6),(5, 5),(5, 6),(6, 4), (6, 5), (6, 6)}
F = {(1, 5),(2, 5),(3, 5),(4, 5),(5, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6)}

$$\begin{gathered} \mathrm{Now}, \\ E\cap F=\left\{\left(5,5\right),\left(5,6\right),\left(6,5\right)\right\} \\ \therefore \mathrm{Required}\mathrm{probability}=P\left(E/F\right)=\frac{n\left(E\cap F\right)}{n\left(F\right)}=\frac{3}{11} \end{gathered}$$