Question

The probability of drawing two aces from a standard deck is 0.0059. We know this probability, but we don't know if the first card was replaced. If the two draws are defined as event A and event B, are the events dependent or independent?.

Answer 1

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:

$\dfrac{1}{13}\cdot \dfrac{1}{17} =\dfrac{1}{221}\approx 0.0045.$

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:

$\dfrac{1}{13}\cdot \dfrac{1}{13} =\dfrac{1}{169}\approx 0.0059.$

3. If events A and B are independent, then $Pr(A\cap B)=Pr(A)\cdot Pr(B).$

All these three steps show you that the first card was replaced and events are independent.

Answer 2

If the two draws are defined as event A and event B, then they are independent because, based on the probability, the first ace was replaced before drawing the second ace.

 

Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

 

I am hoping that this answer has satisfied your query and it will be able to help you in your endeavor, and if you would like, feel free to ask another question.