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

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.

Answer 2

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:

`(1)/(13)\cdot (1)/(17) =(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:

`(1)/(13)\cdot (1)/(13) =(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.