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)}

\[\text{ Now } , \]

\[E \cap F = \left\{ \left( 5, 5 \right), \left( 5, 6 \right) \right\}\]

\[ \therefore \text{ Required probability }  = P\left( E/F \right) = \frac{n\left( E \cap F \right)}{n\left( F \right)} = \frac{2}{6} = \frac{1}{3}\]

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)}

\[\text{ Now } , \]

\[E \cap F = \left\{ \left( 5, 5 \right), \left( 5, 6 \right), \left( 6, 5 \right) \right\}\]

\[ \therefore \text{ Required probability } = P\left( E/F \right) = \frac{n\left( E \cap F \right)}{n\left( F \right)} = \frac{3}{11}\]