Question

Consider a standard deck of cards. Consider A,K,Q,J as face cards; the others are even or odd. What is the probability that a card drawn at random is a. A king if it is black b. A king if it is a face card c. an even card if it is not a face card d. a face card if it not an even card e. the king of spades assuming it is red f. a black card given that it is a spade g. a spade if it is black h. a red face card i. a jack of diamonds j. a jack or a diamond

Answer 1

Total number of cards = 52

Number of face cards = 4*4 = 16

Number of non face cards = 52 - 16 = 36

Total number of black cards = 26

Total number of red cards = 26

a.

P[ A king if it is black ] = P[ A black king ] / P[  A black card ]

P[ A black king ] = 2/52 ( 2 black king and two red king )

P[  A black card ] = 26/52 ( 26 black card in total )

P[ A king if it is black ] = (2/52)/(26/52)

P[ A king if it is black ] = 2/26

P[ A king if it is black ] = 1/13

b.

P[ A king if it is a face card ] = P[ A king ] / P[  A face card ]

P[ A king if it is a face card ] = 4/52 ( 4 kings in total )

P[  A king if it is a face card ] = 16/52 ( 16 face cards in total )

P[ A king if it is a face card] = (4/52)/(16/52)

P[ A king if it is a face card ] = 4/16

P[ A king if it is black ] = 1/4

c.

P[ an even card if it is not a face card ] = P[ an even card and non face card ] / P[ it is not a face card ]

P[ an even card and non face card ] = 20/52 ( 16 are odd and 20 are even )

P[ it is not a face card ] = 36/52

P[ an even card if it is not a face card ] = ((30/52)/(36/52))

P[ an even card if it is not a face card ] = 30/36

P[ an even card if it is not a face card ] = 5/6

d.

P[ a face card if it not an even card ] = P[ an odd card and face card ] / P[ an odd card ]

P[ an odd card ] = 28/52 ( 28 are odd in total and 24 are even )

P[ an odd card and face card ] = 12/52 ( 12/ are odd in and 4 are even )

P[ a face card if it not an even card ] = ((12/52)/(28/52))

P[ a face card if it not an even card ] = 12/28

P[ a face card if it not an even card ] = 3/7

e.

P[ the king of spades assuming it is red ] = P[ the king of spades and it is red ] / P[ it is red ]

P[ the king of spades and it is red ] = 0 ( spade is black )

P[ the king of spades assuming it is red ] = 0

f.

P[ a black card given that it is a spade ] = P[ a black card given and it is a spade ]/ P[ it is a spade ]

P[ it is a spade ] = 13/52

P[ a black card given and it is a spade ] = 13/52 ( only 13 spade in total )

P[ a black card given that it is a spade ] = ((13/52)/(13/52))

P[ a black card given that it is a spade ] = 1

g.

P[ a spade if it is black ] = P[ a spade and black ] / P[ it is black ]

P[ it is black ] = 26/52

P[ a spade and black ] = 13/52

P[ a spade if it is black ] = ((13/52)/(26/52))

P[ a spade if it is black ] = 13/26

P[ a spade if it is black ] = 1/2

h.

P[ a red face card ] = Number of red face cards / number of total cards

Number of red face cards = 8

number of total cards = 52

P[ a red face card ] = 8/52

P[ a red face card ] = 2/13

i.

P[ a jack of diamonds ] = number of jack of diamonds / number of total cards

number of jack of diamonds = 1

number of total cards = 52

P[ a jack of diamonds ] = 1/52

j.

P[ a jack or a diamond ] = P[ a jack ] + P[ a diamond ] - P[ a jack of diamonds ]

P[ a jack ] = 4/52 ( 4 jacks )

P[ a jack ] = 1/13

P[ a diamond ] = 13/52 ( 13 diamonds )

P[ a diamond ] = 1/4

P[ a jack of diamonds ] = 1/52 ( from previous part )

P[ a jack or a diamond ] = ( 4/52 ) + ( 13/52 ) - (1/52)

P[ a jack or a diamond ] = 16/52

P[ a jack or a diamond ] = 4/13