Answer :
Given: A 6-sided die is given.
Required: To determine the probability of getting a 2, given you know the number is even.
Explanation: The problem is based on conditional probability.
The probability of an event X given that event Y has happened is-
[tex]P(\frac{X}{Y})=\frac{P(X\cap Y)}{P(Y)}[/tex]Here, Let A be the event that outcome is an even number.
Then,
[tex]\begin{gathered} A=\lbrace2,4,6\rbrace \\ P(A)=\frac{3}{6} \end{gathered}[/tex]Let B be the event of getting 2.
Then,
[tex]\begin{gathered} B=\lbrace2\rbrace \\ P(B)=\frac{1}{6} \end{gathered}[/tex]Now,
[tex]\begin{gathered} A\cap B=\lbrace2\rbrace \\ P(A\cap B)=\frac{1}{6} \end{gathered}[/tex]Hence the probability of getting a 2, given you know the number is even, is-
[tex]\begin{gathered} P(\frac{B}{A})=\frac{P(A\cap B)}{P(A)} \\ =\frac{\frac{1}{6}}{\frac{3}{6}} \\ =\frac{1}{3} \end{gathered}[/tex]Final Answer: The probability of getting a 2, given you know the number is even, is-
[tex]\frac{1}{3}[/tex]