Question

You roll one 6-sided die. What is the probability of getting a 2 given you know the number is even?2313121

Answer 1

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-

$P(\frac{X}{Y})=\frac{P(X\cap Y)}{P(Y)}$

Here, Let A be the event that outcome is an even number.

Then,

$\begin{gathered} A=\lbrace2,4,6\rbrace \\ P(A)=\frac{3}{6} \end{gathered}$

Let B be the event of getting 2.

Then,

$\begin{gathered} B=\lbrace2\rbrace \\ P(B)=\frac{1}{6} \end{gathered}$

Now,

$\begin{gathered} A\cap B=\lbrace2\rbrace \\ P(A\cap B)=\frac{1}{6} \end{gathered}$

Hence the probability of getting a 2, given you know the number is even, is-

$\begin{gathered} P(\frac{B}{A})=\frac{P(A\cap B)}{P(A)} \\ =\frac{\frac{1}{6}}{\frac{3}{6}} \\ =\frac{1}{3} \end{gathered}$

Final Answer: The probability of getting a 2, given you know the number is even, is-

$\frac{1}{3}$