Probability, dependent Events I am trying to understand some of the tasks and would be happy if someone could check my solution and possibly point out any mistakes.
In advance, I am not asking for a solution, because I would like to work it out by myself.
The purpose of this exercise is to check whether events A and B are independent of each other.
Three dices are thrown.
The events are defined as follows:
All dice have the same number of dots.
B: The total sum of the dices is less than 5.
Therefore is
$A = \left \{ \left ( 1,1,1 \right ), (2,2,2), (3,3,3), (4,4,4), (5,5,5), (6,6,6) \right \}$
$B = \left \{ \left ( 1,1,1 \right ), (1,1,2), (1,2,1), (2,1,1) \right \}$
Some intermediate results:
$P\left ( A \right ) = \frac{6}{216} = \frac{3}{108}$
$P\left ( not A \right ) = 1 - P(A) = 1 - \frac{3}{108} = \frac{35}{36}$
$P(B) = \frac{4}{216} = \frac{1}{54}$
$P(A\cap B) = \frac{1}{216}$
$P(notA\cap B) = \frac{3}{216}$
A and B are independent if
$P(B|notA) = P(B|A)$
$
P(B|A) = \frac{P(A\cap B)}{P(A)} = \frac{\frac{1}{216}}{\frac{6}{216}} = \frac{1}{6}
$
$
P(B|notA) = \frac{P(notA\cap B)}{P(notA)} = \frac{\frac{3}{216}}{\frac{35}{36}} = \frac{1}{70}
$
Thus, the events A and B are dependent.
Thank you in advance for your time.
