Task 1: Independence a) Consider a regular deck of playing cards and the two eve

Question

Task 1: Independence a) Consider a regular deck of playing cards and the two events "card drawn is a queen" and "card drawn is a heart." Suppose I shuffle the deck, randomly draw one card, and, before looking at the card, I ask you what is the probability that it is a "queen". What is your answer? b) Then I peek at the card and tell you that it is a "heart."Now, what is the probability that the card is a "queen"? c) How does this answer compare to the previous? Did the additional information that the card was a heart change the probability that it was a queen? d) What can we conclude about the events "card drawn is a queen" and "card drawn is a heart" e) Furthermore, suppose that after I drew the card and looked at it, I had told you the card was not a heart". What would be the probability the card is a queen? f) Did this additional information change the probability that it was a "queen"? What can you conclude now about the dependence or independence of these three events? In equation form: P(queen Icard is a heart) P (Q IH) P (Q) P(queen lcard is not a heart) P (Q Inot H)- P (Q) Therefore, P(Q) P(QIH) P(Qlnot H), and the events are (independent/dependent?)

Explanation / Answer

a) Total number of possible outcomes of randomly picking up a card from a deck of 52 cards = 52C1

Total Number of favourable outcomes (that the randomly picked card is a queen) = 4C1

Probability=4C1/52C1=4/52=1/13

b) Total number of possible outcomes when I know that the card drawn is a heart=13C1

Total number of favourable outcomes (that the randomly picked card is a queen when I know that the picked card is a heart) = 1

Probability that the card is a queen given that the card is a heart = 1/13

c) No, this does not change the probability as it is 1/13 in both cases. The reason for this that, both are equally likely event. Choosing 1 Queen out of 52 cards which have 4 queens in them and choosing 1 Queen out of 13 cards which has 1 queen in it. Notice the ratio remains the same.

d) Probability that the card drawn is a queen (as calculated in the first part) = 1/13

Probability that the card drawn is a heart = (Total number of hearts available)/(Total no. of cards in the deck)

=13/52=1/4

Also, P(card drawn is Queen|card drawn is heart)

= P(card drawn is queen and card drawn is heart)/P(card drawn is heart)

=(1/52)/(13/52)=1/13

and P(Card drawn is a queen) = 13/52 = 1/13

So the event of card being drawn is a queen is not dependent on the event of the card drawnn being heart.

e) P(queen | card is not a heart) = P(queen and card is not a heart)/P(card is not a heart)

=(3/52)/(39/52)=1/13

f) The events are independent as P(Queen)=P(Queen|Hearts)=P(Queen|Not Hearts)=1/13

So, they are independent events.

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