Question

A standard deck of cards contains 54 cards: 4 suits (spades, clubs, hearts, and diamonds) with...

A standard deck of cards contains 54 cards: 4 suits (spades, clubs, hearts, and diamonds) with 13 cards (2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, and an ace) in each suit as well as two jokers. Half of the cards are red, and the other half of the cards are black (this includes jokers where one is black and the other is red). Using this information answer the following questions.

1. List the sample space of even numbered red cards.

2. List the sample space of black face cards (jack, queen, king, and ace).

3. Assuming these rules: number cards are equal to their face value, face cards are worth 10, and jokers are wild and can be any value from 2-10. List the sample space where two cards numerical total equals 17 (cards are replaced).

4. Assuming these rules: number cards are equal to their face value, face cards are worth 10, and jokers are wild and can be any value from 2-10. List the sample space where two cards numerical total equals 17 (cards are NOT replaced).

5. Calculate the probabilities of the above questions assuming one standard deck of 54 cards.

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Answer #1

A standard deck of cards contains 54 cards: 4 suits (spades, clubs, hearts, and diamonds) with 13 cards (2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, and an ace) in each suit as well as two jokers. Half of the cards are red, and the other half of the cards are black (this includes jokers where one is black and the other is red).

1)2,4,6,8,10 are even numbered cards.

The sample space of even numbered red cards:

   = { 2 of heart ,2 of diamond ,4 of heart ,4 of diamond, 6 of heart, 6 of diamond ,8 of heart , 8 of diamond,10 of heart ,10 of diamond}

2)There are 2 black jack, 2 black queen, 2 black king, 2 black ace.

The sample space of black face cards (jack, queen, king, and ace)

   = { jack of club , jack of spade, queen of club, queen of spade, king of club, king of spade, ace of spade, ace of club}

3)Number cards are equal to their face value, face cards are worth 10, and jokers are wild and can be any value from 2-10.

i.e.

Value of 2 is 2, 3 is 3 ,4 is 4, and so on.

Face cards = 10.

And joker can take any value from 2 to 10 .

Sun must be 17

the sample space where two cards numerical total equals 17

= { [2*( 7 of spade, 10 of spade), 2*(7 of spade, 10 of heart) ,2*( 7 of spade, 10 of diamond), 2*( 7 of spade, 10 of club),2* ( 7 of spade, jack of spade), 2*(7 of spade, jack of club) , 2*( 7 of spade, jack of heart) ,2* ( 7 of spade, jack of diamond),2* ( 7 of spade, queen of spade), 2*( 7 of spade, queen of club), 2*( 7 of spade, queen of heart ) ,2*(7 of spade, queen of diamond),2*( 7 of spade, ace of club) ,2*( 7 of spade, ace of heart) ,2* ( 7 of spade, ace of spade), 2*( 7 of spade, ace of diamond), 2*(7 of spade, king of spade), 2*(7 of spade, king of club), 2*(7 of spade, king of heart) ,2*(7 of spade, king of diamond), 4* ( 7 of spade, joker (if =10))],.....[2*( 8 of spade, 9 of spade),2*(8 of spade, 9 of club),2* ( 8 of spade, 9 of diamond), 2*(8 of spade, 9 of heart)],...}

Same continues for 7 of club, 7 of diamond, 7 of heart and also for 8 of club, 8 of diamond, 8 of heart. As this is with replacement each event occurs 2 times.

4)the sample space where two cards numerical total equals 17,

={[( 7 of spade, 10 of spade), (7 of spade, 10 of heart) ,( 7 of spade, 10 of diamond), ( 7 of spade, 10 of club), ( 7 of spade, jack of spade), (7 of spade, jack of club) , ( 7 of spade, jack of heart) , ( 7 of spade, jack of diamond), ( 7 of spade, queen of spade), ( 7 of spade, queen of club), ( 7 of spade, queen of heart ) ,(7 of spade, queen of diamond),( 7 of spade, ace of club) ,( 7 of spade, ace of heart) , ( 7 of spade, ace of spade), ( 7 of spade, ace of diamond), (7 of spade, king of spade), (7 of spade, king of club), ( 7 of spade, king of heart), (7 of spade, king of diamond), ( 7 of spade, red joker ( if =10)), ( 7 of spade ,black joker (if = 10)),..., ( 8 of spade, 9 of spade),(8 of spade, 9 of club), ( 8 of spade, 9 of diamond), (8 of spade, 9 of heart),...}

Same continues for 7 of club, diamond, heart. And also for 8 of club, diamond, heart.

5) The probabilities are,

1) the probability of selecting even numbered red cards is = 10/54 = 0.1851

2) Probability of selecting black face cards are

= 8/54 = 0.1481

3) Probability of selecting 2 cards numerical sum is 17 with replacement

=[ (44*4)+(8*4)] / 542

= [ 208/2916] = 0.0713

4) Probability of selecting 2 cards numerical sum is 17 without replacement

= 104/1431 = 0.0727

***** If you have any queries or doubts please comment below, if you're satisfied please give a like. Thank you!

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