Question

You have a well-shuffled deck of 52 cards. See the diagram of the full deck for reference.

1. One card is chosen from the full deck with replacement, and then a second card is drawn.
Fill in the blanks below to determine the probability that the 1st card is a Red 8 and the second card is a Black King.

P(1st pick is a Red 8) =  
P(2nd pick is a Black King) =
P(1st is a Red 8 and 2nd is a Black King) =  


2. One card is chosen from the full deck without replacement, and then a second card is drawn.
Fill in the blanks below and determine the probability that the 1st card is a Red 8 and the second card is a Black King.
P(1st pick is a Red 8) =  
P(2nd pick is a Black King | 1st pick is a Red 8) =
P(1st is a Red 8 and 2nd is a Black King) =

Answer 1

Given, a deck of 52 cards.

This deck of cards contain : 26 cards of red, 26 cards od black, 4 cards of each (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K) with 2 in black and 2 in red.

Now the experiment shows that one card is chosen from the full deck which is a Red 8 and a second card is drawn which is a Black King.

1. Case: With Replacement of Red 8 card

Here, Probability of an event = Possibility of occurence of an event / Total outcomes of an event

a. Probability that the 1st pick is Red 8 is asked to be calculated.

In this case, Possibility of occurence of Red 8 = 2 ( as there are 2 red cards with 8 on it )

Total outcomes of this event = 52 ( as there are 52 cards in the deck and any card can be picked up )

Therefore, Probability ( Red 8 ) = 2 / 52 = 1 / 26

b. As this case is dealing with the replacement i.e., after drawing the red 8 card it is again replaced into the deck of cards. Now, there are again 52 cards in the deck.

Probability of occurence of Black King With Replacement of Red 8 card is to be calculated.

In this case, Possibility of Occurence of Black King with Replacement of Red 8 card= 2 ( as there exists 2 Black Kings in the deck )

Total outcomes of this event = 52 ( as there are 52 cards in the deck and any card can be picked up )

There, Probability ( Black King with Replacement of Red 8 card ) = 2 / 52 = 1 / 26

c. Probability ( Red 8 and Black King with Replacement of Red 8 card ) is to be calculated.

These two events are mutually exclusive events i.e., the occurence of one event doesn't affect the occurence of another event.

This makes Probability ( Red 8 and Black King with Replacement of Red 8 card ) = Probability ( Red 8 ) * Probability ( Black King with Replacement of Red 8 card )

Now, substitute the values calculated in the above points.

Therefore, Probability ( Red 8 and Black King With Replacement of Red 8 card ) = ( 1 / 26 ) * ( 1 / 26 ) = 1 / 676

2. Case: Without Replacement of Red 8 card

a. Probability that the 1st pick is Red 8 is asked to be calculated.

In this case, Possibility of occurence of Red 8 = 2 ( as there are 2 red cards with 8 on it )

Total outcomes of this event = 52 ( as there are 52 cards in the deck and any card can be picked up )

Therefore, Probability ( Red 8 ) = 2 / 52 = 1 / 26

b. As this case is dealing without replacement i.e., after drawing the red 8 card it is not replaced into the deck of cards. Therefore, the deck of cards contain only 51 cards ( 52 - 1) as Red 8 card is not replaced and the Black King is drawn.

Probability of occurence of Black King Without Replacement of Red 8 card is to be calculated.

In this case, Possibility of Occurence of Black King Without Replacement of Red 8 = 2 ( as there exists 2 Black Kings in the deck )

Total outcomes of this event = 51 ( as there are 51 cards in the deck and any card can be picked up )

There, Probability ( Black King Without Replacement of Red 8 card) = 2 / 51

c. Probability ( Red 8 and Black King Without Replacement of Red 8 card ) is to be calculated.

These two events are mutually exclusive events i.e., the occurence of one event doesn't affect the occurence of another event.

This makes Probability ( Red 8 and Black King Without Replacement of Red 8 card ) = Probability ( Red 8 ) * Probability ( Black King Without Replacement of Red 8 card )

Now, substitute the values calculated in the above points.

Therefore, Probability ( Red 8 and Black King of Red 8 card ) = ( 1 / 26 ) * ( 2 / 51 ) = 2 / 1326 = 1 / 663