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I am not understanding well the available answers for this question “Going back to the card example, which of the following sets of events are independent?”

I am not understanding well the available answers for this question “Going back to the card example, which of the following sets of events are independent?”

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The answer given to this question “Going back to the card example, which of the following sets of events are independent?” was “Drawing a diamond and drawing an ace”. Why?
How are these two events independent? There is an Ace-Diamond, right?
Why are these two events “drawing a queen and drawing a Jack” dependent?
Why are these two events “drawing a four and an ace of spades” dependent?
The only events that were clearly dependent were drawing a heart and drawing the Jack of hearts. The other answers were pretty confusing.
Please help me out. Thanks a lot.
 

2 Answers

365 Team

Hey Luis,

Thanks for reaching out!

So, in this case we’re examining the events in pairs (i.e. A: Draw a Diamond, B: Draw an Ace).

We can state that two events are independent if the conditional and unconditional probabilities are equal (P(A|B ) = P(A)). Hence, if the likelihood of drawing a Diamond (P(A)) is the same as drawing a diamond, provided we knew the card was an Ace (P(A|B )), then the two are independent. In this specific case, P(A) = 1/4 since there are 13 diamonds in the 52-card deck. Simultaneously, there is a single diamond among the 4 aces in the deck, so P(A|B ) = 1/4.

Hence, P(A) = 1/4 =  P(A|B ), so the two are independent.

We can try the same for the other pairs and see that’s not the case, so this is the only acceptable answer.

Hope this helps!
Best,
365 Vik

P.S. We’re drawing a single card.

Hi there, looking at this example – drawing a heart and drawing a Jack of hearts would it be correct to say p(A) – Drawing a heart then P(A) = 1/4, P(A|B) = Drawing a heart given that we have drawn a Jack of heart = 1. Therefore the two events are not independent?

1 week

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Hi Vik,
 
I’m trying to understand the statement when P(A|B) = P(A), then those two events are independent. Could you please elaborate more? Thanks!

Hi Anindio. Try to think of it this way. Given that you have 2 events A and B, if the probability of getting a favorable outcome for one of them (let’s say A) P(A) is unaffected by a previous favorable outcome of the other (B in this case) P(A|B), then they are independent, as their outcomes are independent of each other. An example would be: A = getting heads on a coin flip; B = getting tails on a coin flip. Does the probability of getting a heads, P(A) change if we previously got a tails? No, right? This means that P(A|B) = P(A) = 1/2 = 0.5. Because P(A) = P(A|B), we can say events A and B are independent. Hope this helps!

2 days