INDEPENDENT AND DEPENDENT EVENTS

Question

Consider the experiment of randomly selecting a one card from a deck of 52 playing cards

a.) Determine whether the event of selecting a king is independent of the event selecting a face.

b.) Determine whether the event of selecting a king is independent of the event selecting a heart.

Answer 1

We can consider what we know the results should be, then confirm that through computation:

Our informal understanding of "independent" events is that knowing whether one event happened does not change the probability that the other happened also. For example, if we could pick a random high school student from a high school population and look at 2 distinct events: A) the student we picked was from grade 9 and B) the student we picked had blue eyes. We would expect these to be independent. Knowing the grade level of the student wouldn't change the likelihood they had blue eyes. However, we could do the same experiment, picking a random hs student and look at the probabilities of these two events: A) the student we picked was from grade 9 and B) the student we picked had a driver's license. In this case, we would expect these to be dependent: in other words, knowing the student was from grade 9 would make the likelihood they held a license go down significantly.

Moving on to cards, we expect the events in a) to be dependent: if we know we picked a face card, the odds that the card we picked was a king go up significantly. However, the events in b) will be independent: knowing we picked a heart does not change the likelihood we have picked a king.

We define independence by using the concept of a conditional probability: P(B|A), which reads as "the probability of B given A." If P(B|A) = P(B), then A and B are said to be independent.

In a) let K: we picked a King and let F: we picked a face card.

P(K) = 1/13 since there are 4 kings in a 52-card deck. But P(K|F) = 1/3 since there are 4 kings out of 12 face cards. And P(K|F) is the probability we have picked a king knowing it is true we picked a face card. We conclude that P(K) ≠ P(K|F) and that K and F are dependent events.

We reach the opposite conclusion in b). Let H: we picked a heart. Now P(K) = 1/13 still, and P(K|H) = 1/13 too! There are 13 hearts in the deck, exactly one of which is a king. P(K) = P(K|H) and K and H are independent.

Lastly, there is a different way to determine independence that is equally valid: If P(A AND B) = P(A)·P(B) then the events are independent. I will leave it to you to show that such is the case in b) but not so in a).