# Thread: Are these events independent?

1. ## Are these events independent?

Choose a number X from {1,2,3,4,5}. Now choose a number from the subset {1.....X}. Call this second number Y.

P{X|Y=1} = 1/5
P{X|Y=2| = 1/4
P{X|Y=3} = 1/3
P{X|Y=4} = 1/2
P{X|Y=5} = 1

Are these probabilities correct? These event are not independent?

2. Originally Posted by lord12
Choose a number X from {1,2,3,4,5}. Now choose a number from the subset {1.....X}. Call this second number Y.

P{X|Y=1} = 1/5
P{X|Y=2| = 1/4
P{X|Y=3} = 1/3
P{X|Y=4} = 1/2
P{X|Y=5} = 1

Are these probabilities correct? These event are not independent?
No its P(X|Y=1) = 1

P(X|Y=2) = 4/5

P(X|Y=3) = 3/5

etc...

still not independent because $P(X|Y) \neq P(X)$.

3. Originally Posted by particlejohn
No its P(X|Y=1) = 1

P(X|Y=2) = 4/5

P(X|Y=3) = 3/5

etc...

still not independent because $P(X|Y) \neq P(X)$.
$\Pr(X = 1 | Y = 1) \neq \Pr(X = 1)$ so X and Y are clearly dependent.

I'm not sure what Pr(X | Y = 1) = 1 means ......

$\Pr(X = 1 | Y = 1) = \frac{ \Pr(Y = 1 | X = 1) \cdot \Pr(X = 1) }{\Pr(Y = 1)} = \frac{ \frac{1}{5}}{\Pr(Y = 1)}$

where

$\Pr(Y = 1) = \Pr(Y = 1 | X = 1) \cdot \Pr(X = 1)$ $+ \Pr(Y = 1 | X = 2) \cdot \Pr(X = 2) + \Pr(Y = 1 | X = 3) \cdot \Pr(X = 3)$

$+ \Pr(Y = 1 | X = 4) \cdot \Pr(X = 4) + \Pr(Y = 1 | X = 5) \cdot \Pr(X = 5)$

$= (1) \, \left(\frac{1}{5}\right) + \left(\frac{1}{2}\right) \cdot \left(\frac{1}{5}\right)$ $+ \left(\frac{1}{3}\right) \cdot \left(\frac{1}{5}\right) + \left(\frac{1}{4}\right) \cdot \left(\frac{1}{5}\right)$ $\left(\frac{1}{5}\right) \cdot \left(\frac{1}{5}\right)$

$= \frac{1}{5} + \frac{1}{10} + \frac{1}{15} + \frac{1}{20} + \frac{1}{25} = \frac{137}{300}$.

Therefore
$\Pr(X = 1 | Y = 1) = \frac{ \frac{1}{5}}{\frac{137}{300}} = \frac{60}{137}$.

Obviously $\Pr(X = 2 | Y = 1) = \frac{ \frac{1}{10}}{\frac{137}{300}} = \frac{30}{137}$.

And so on .....

Does Pr(X | Y = 1) mean Pr(X = 1, 2, 3, 4 or 5 | Y = 1) .....?