A standard deck of playing cards consists of 52 playing cards.
1. Count the probability of drawing two aces from a standard deck without replacment.
Among 52 playing cards are 4 aces, then the probability to select first ace is 4/52=1/13. After picking out first ace, only 3 aces left and in total 51 playing cards left, then the probability to select second ace is 3/51=1/17. Use the product rule to find the probability to select two aces without replacement:
[tex] \dfrac{1}{13}\cdot \dfrac{1}{17} =\dfrac{1}{221}\approx 0.0045. [/tex]
2. Count the probability of drawing two aces from a standard deck with replacment.
Among 52 playing cards are 4 aces, then the probability to select first ace is 4/52=1/13. After picking out first ace, this card was returned back into the deck and the probability to select second ace is 4/52=1/13 too. Use the product rule to find the probability to select two aces with replacement:
[tex] \dfrac{1}{13}\cdot \dfrac{1}{13} =\dfrac{1}{169}\approx 0.0059. [/tex]
3. If events A and B are independent, then [tex] Pr(A\cap B)=Pr(A)\cdot Pr(B). [/tex]
All these three steps show you that the first card was replaced and events are independent.
