Example: Consider the following: two cards are randomly taken out of the deck in

Question

Example: Consider the following: two cards are randomly taken out of the deck in the tollowing way: after the first card is taken out and observed, it is placed back into the deck and the deck is shuffled prior to the draw of the second card. i. If the first card taken out of the deck is a e, what is the chance that the second card taken out is a ? ii. What is the probability of getting 2 e's in a row? ii. What is the probability of getting 3's in a row? (b) Suppose the cards are chosen in such a way that after the first card is taken out and observed, it is not placed back into the deck and the deck is shuffled prior to the draw of the second card. Repcat (a).

Explanation / Answer

Solution:

      Two cards are randomly chossen with replacement.

Part a) i) If first cards drawn is Spade cards , what is the proabability that second card is spade?

that is we have to find : P( Second Spade card given that first card is spade)=..............?

Since two cards are drawn with replacement, drawing of first card does not affect on drawing of second card.

There are 13 spade cards out of 52 playing cards.

Thus, P( Second Spade card given that first card is spade)=13/52 = 1/4=0.25

ii) What is the probability of getting 2 spade in a row ?

That is : P( First is Spade and Second is spade) = P( First Spade card) x P( Second Spade card)

             P( First is Spade and Second is spade) = 13/52 x 13/52

             P( First is Spade and Second is spade) = 1/4 x 1/4

             P( First is Spade and Second is spade) = 0.25 x 0.25

             P( First is Spade and Second is spade) = 0.0625

iii) What is the probability of getting 3 spade in a row ?

Since we are drawing only two cards from deck of 52 playing cards, getting 3 spade cards is an impossible event,

and probability of an impossible event is 0.

Hence probability getting 3 spade cards in a row is 0

Part b) Two cards are drawn without replacement.

     i) If first card is spade , then what is the probability of getting second spade card?

that is we have to find : P( Second Spade card given that first card is spade)=..............?

Since for first draw , we know that the card is spade, and it is not replaced back to the deck. hence we have total 12 spade cards and total 51 cards in that deck.

Hence P( Second Spade card given that first card is spade)= 12/51 =0.2353

ii) What is the probability of getting 2 spade in a row ?

That is : P( First is Spade and Second is spade) = P( First Spade card) x P( Second Spade card)

             P( First is Spade and Second is spade) = 13/52 x 12/51

             P( First is Spade and Second is spade) = 0.25 x 0.2353

             P( First is Spade and Second is spade) = 0.0588

iii) What is the probability of getting 3 spade in a row ?

Since we are drawing only two cards from deck of 52 playing cards, getting 3 spade cards is an impossible event,

and probability of an impossible event is 0.

Hence probability getting 3 spade cards in a row is 0

   

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