a.
To obtain: The probability of getting a King or a queen or a jack.
a.
Answer to Problem 23E
The probability of getting a King or a queen or a jack is
Explanation of Solution
Given info:
One card is drawn from an ordinary deck of cards.
Calculation:
In ordinary deck of cards there are 4 suits. They are hearts, clubs, diamonds and spades. In each suite there are 13 cards. In 13 cards, nine cards are numbers from 2 to 10 and remaining four cards are king, queen, ace and jack cards.
Here, the cards are a King or a queen or a jack.
The total number of outcomes is 52.
Let
The number of elements in A is 4.
The formula for probability of event A is,
Substitute 4 for ‘Number of outcomes in A’ and 52 for ‘Total number of outcomes’,
Let event B denote that getting a queen.
The number of elements in B is 4.
The formula for probability of event B is,
Substitute 4 for ‘Number of outcomes in B’ and 52 for ‘Total number of outcomes’,
Let event C denote that getting jack.
The number elements in C is 4.
The formula for probability of event C is,
Substitute 4 for ‘Number of outcomes in C’ and 52 for ‘Total number of outcomes’,
The formula for probability of getting A or B or C is,
Substitute
Thus, the probability of getting a King or a queen or a jack is
b.
To obtain: The probability of getting a club or a heart or a spade.
b.
Answer to Problem 23E
The probability of getting a club or a heart or a spade is
Explanation of Solution
Calculation:
Let event A denote that the outcome is club, event B denote that the outcome is heart and event C denote that the outcome is a spade.
The formula for probability of getting A or B or C is,
The number of club cards is 13.
The total number of outcomes is 52.
The formula for probability of event A is,
Substitute 13 for ‘Number of outcomes in A’ and 52 for ‘Total number of outcomes’,
The number of heart cards is 13.
The total number of outcomes is 52.
The formula for probability of event B is,
Substitute 13 for ‘Number of outcomes in B’ and 52 for ‘Total number of outcomes’,
The number of spade cards is 13.
The total number of outcomes is 52.
The formula for probability of event C is,
Substitute 13 for ‘Number of outcomes in C’ and 52 for ‘Total number of’,
Substitute
Thus, the probability of getting a club or a heart or a spade is
c.
To obtain: The probability of getting a King or a queen or a diamond.
c.
Answer to Problem 23E
The probability of getting a King or a queen or a diamond is
Explanation of Solution
Calculation:
In ordinary deck of cards there are 4 suits. They are hearts, clubs, diamonds and spades. In each suite there are 13 cards. In 13 cards, nine cards are numbers from 2 to 10 and remaining four cards are king, queen, ace and jack cards.
Here, the cards are a King or a queen or a diamond.
The total number of outcomes is 52.
Let event A denote that getting a King.
The number of elements in A is 4.
The formula for probability of event A is,
Substitute 4 for ‘Number of outcomes in A’ and 52 for ‘Total number of outcomes’,
Let event B denote that getting a queen.
The number of elements in B is 4.
The formula for probability of event B is,
Substitute 4 for ‘Number of outcomes in B’ and 52 for ‘Total number of outcomes’,
Let event C denote that getting diamond.
The number of diamond cards is 13.
The total number of outcomes is 52.
The formula for probability of event C is,
Substitute 13 for ‘Number of outcomes in C’ and 52 for ‘Total number of’,
Events A and B cannot be occur together. Therefore
The formula for probability of event A and C is,
Substitute 1 for ‘Number of outcomes in A and C’ and 52 for ‘Total number of outcomes’,
The formula for probability of event B and C is,
Substitute 1 for ‘Number of outcomes in B and C’ and 52 for ‘Total number of outcomes’,
There is no element is common to the events A, B, and C. Therefore
The formula for probability of getting A or B or C is,
Substitute
Thus, the probability of getting a King or a queen or a diamond is
d.
To obtain: The probability of getting an ace or a diamond or a heart.
d.
Answer to Problem 23E
The probability of getting an ace or a diamond or a heart is
Explanation of Solution
Calculation:
Let event A denote that the outcome is ace, event B denote that the outcome is diamond and event C denote that the outcome is club.
The formula for probability of getting A or B or C is,
The number of ace cards is 4.
The total number of outcomes is 52.
The formula for probability of event A is,
Substitute 4 for ‘Number of outcomes in A’ and 52 for ‘Total number of outcomes’,
The number of diamond cards is 13.
The total number of outcomes is 52.
The formula for probability of event B is,
Substitute 13 for ‘Number of outcomes in B’ and 52 for ‘Total number of outcomes’,
The number of heart cards is 13.
The total number of outcomes is 52.
The formula for probability of event C is,
Substitute 13 for ‘Number of outcomes in C’ and 52 for ‘Total number of outcomes in the
The formula for probability of event A and B is,
Substitute 1 for ‘Number of outcomes in A and B’ and 52 for ‘Total number of outcomes in the sample space’,
The formula for probability of event A and C is,
Substitute 1 for ‘Number of outcomes in A and C’ and 52 for ‘Total number of outcomes in the sample space’,
There is no intersection part for events B and C. Hence
Substitute
Thus, the probability of getting an ace or a diamond or a heart is
e.
To obtain: The probability of getting a 9 or a 10 or a spade or a club.
e.
Answer to Problem 23E
The probability of getting a 9 or a 10 or a spade or a club is
Explanation of Solution
Calculation:
Let event A denote that the outcome is a 9, event B denote that the outcome is a 10, event C denote that the outcome is a spade and D denote that the outcome is a club.
The formula for probability of getting A or B or C or D is,
The number of 9 cards is 4.
The total number of outcomes is 52.
The formula for probability of event A is,
Substitute 4 for ‘Number of outcomes in A’ and 52 for ‘Total number of outcomes’,
The number of 10 cards is 4.
The total number of outcomes is 52.
The formula for probability of event B is,
Substitute 4 for ‘Number of outcomes in B’ and 52 for ‘Total number of outcomes’,
The number of spade cards is 13.
The total number of outcomes is 52.
The formula for probability of event C is,
Substitute 13 for ‘Number of outcomes in C’ and 52 for ‘Total number of outcomes’,
The number of club cards is 13.
The total number of outcomes is 52.
The formula for probability of event D is,
Substitute 13 for ‘Number of outcomes in D’ and 52 for ‘Total number of outcomes’,
The formula for probability of event A and C is,
Substitute 1 for ‘Number of outcomes inA and B’ and 52 for ‘Total number of outcomes in the sample space’,
The formula for probability of event A and D is,
Substitute 1 for ‘Number of outcomes in A and D’ and 52 for ‘Total number of outcomes in the sample space’,
The formula for probability of event B and C is,
Substitute 1 for ‘Number of outcomes in B and C’ and 52 for ‘Total number of outcomes in the sample space’,
The formula for probability of event B and D is,
Substitute 1 for ‘Number of outcomes inA and D’ and 52 for ‘Total number of outcomes in the sample space’,
There is no intersection part for events A and B, C and D. Hence
Substitute the probabilities of the events in the above equation.
Thus, the probability of getting a 9 or a 10 or a spade or a club is
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Chapter 4 Solutions
Elementary Statistics: A Step-by-Step Approach with Formula Card
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