There are four distinct balls labelled $1, 2, 3, 4$ and four distinct bins labelled A, B, C, D. The balls are picked up in order and placed into one of the four bins at random. Let $E_i$ denote the event that the first $i$ balls go into distinct bins. Calculate the following probabilities. (i) $\Pr[E_4]$ (ii) $\Pr[E_4 \mid E_3]$ (iii) $\Pr[E_4 \mid E_2]$ (iv) $\Pr[E_3 \mid E_4]$. Notation: $\Pr[X] =$ the probability of event $X$ taking place. $\Pr[X \mid Y] =$ the probability of event $X$ taking place, given that event $Y$ has taken place.
There are four distinct balls labelled $1, 2, 3, 4$ and four distinct bins labelled A, B, C, D. The balls are picked up in order and placed into one of the four bins at random. Let $E_i$ denote the event that the first $i$ balls go into distinct bins. Calculate the following probabilities.\\
(i) $\Pr[E_4]$\\
(ii) $\Pr[E_4 \mid E_3]$\\
(iii) $\Pr[E_4 \mid E_2]$\\
(iv) $\Pr[E_3 \mid E_4]$.
Notation: $\Pr[X] =$ the probability of event $X$ taking place. $\Pr[X \mid Y] =$ the probability of event $X$ taking place, given that event $Y$ has taken place.