Answered: In the random experiment consisting of…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

In the random experiment consisting of drawing at random a
card from an ordinary deck of 52 cards, we consider the following events:
A: Draw a black card
B: Draw a queen
C: Draw a heart card
D: Draw a face (jack, queen or king)
Find the probability of the following events:
A, B, C, D, A ∩ B, B ∩ C, A ∩ C, A ∩ B ∩ C, C ∩ D, A ∪ B,
C ∪ D, A ∪ B ∪ C

Expert Solution

Step 1: Probabilities obtained.

In a pack of cards, we have 52 cards having 4 sets of cards

13 Spade (Black Color), 13 Club (Black Color),13 Diamond (Red Color) and 13 Heart (Red color)

A={Black Card) = {Spade ,Club} P(A) = $26 over 52$ =0.5

B={Queen} = {1 card from each 4 sets}, P(B) = $4 over 52$
C= {Heart}, P(C) = $13 over 52$

D={Face cards} = {3 cards from each 4 sets} P(D) = $12 over 52$

$A intersection B$ = {Black Queen 2 cards} , P(A $Error converting from MathML to accessible text.$ B) = $2 over 52$

B $intersection C$ = {1 Queen in Heart), P( $B intersection C$ )= $1 over 52$

$A intersection C$ ={ 0 Heart in Black cards} = P( $ϕ$ )= 0

$A intersection B intersection C$ = { $ϕ$ }, P $left parenthesis A intersection B intersection C right parenthesis$ = 0

C $intersection D$ = {Face cards in Heart}, P( $C intersection D$ ) = $3 over 52$

P(A $union B$ ) = P(A) + P(B) -P( $A intersection B$ ) = $fraction numerator 26 plus 4 minus 2 over denominator 52 end fraction equals 28 over 52$

P(C $union D$ )= P(C) + P(D) -P(C $intersection D$ ) = $fraction numerator 13 plus 12 minus 3 over denominator 52 end fraction equals 22 over 52$

P( $A union B union C$ ) = P(A) + P(B) + P(C) -P $left parenthesis A intersection B right parenthesis space minus space P left parenthesis A intersection C right parenthesis minus space P left parenthesis B intersection C right parenthesis space plus P left parenthesis A intersection B intersection C right parenthesis$

= $fraction numerator 26 plus 4 plus factorial 3 minus 2 minus 0 minus 1 plus 0 over denominator 52 end fraction equals 40 over 52$

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