CALCU-1. The diagram shows the daily Return for Tw

1. The diagram shows the daily Return for Two Stocks Using the Closing Prices. Calculate the covariance of

A B C

and

X Y Z

's return \begin{tabular}{|l|l|l|} \hline Day & \begin{tabular}{l} ABC \\ Returns \end{tabular} & \begin{tabular}{l} XYZ \\ Returns \end{tabular} \\ \hline 1 &

1.10 \%

&

3.00 \%

\\ \hline 2 &

1.70 \%

&

4.20 \%

\\ \hline 3 &

2.10 \%

&

4.90 \%

\\ \hline 4 &

1.40 \%

&

4.10 \%

\\ \hline 5 &

0.20 \%

&

2.50 \%

\\ \hline \end{tabular}

Solution by Steps

step 1

To calculate the covariance of the returns for stocks

ABC

and

XYZ

, we first list the returns for each stock

step 2

The returns for

ABC

are

1.10\%, 1.70\%, 2.10\%, 1.40\%, 0.20\%

and for

XYZ

are

3.00\%, 4.20\%, 4.90\%, 4.10\%, 2.50\%

step 3

We then use the formula for covariance:

\text{Cov}(X,Y) = \frac{1}{n}\sum_{i=1}^{n}(X_i - \bar{X})(Y_i - \bar{Y})

, where

\bar{X}

and

\bar{Y}

are the means of the respective returns

step 4

After calculating the mean of each set of returns and applying the formula, we find the covariance to be

0.665\%

Answer

The covariance of the returns for stocks

ABC

and

XYZ

is

0.665\%

.

Key Concept

Covariance Calculation

Explanation

Covariance measures the degree to which two variables move in relation to each other. In this case, it quantifies the relationship between the returns of two stocks. A positive covariance indicates that the stocks tend to move together.

3. Find the probability of getting a numbered card when a card is drawn from the pack of 52 cards.

Solution by Steps

step 1

Identify the total number of cards in a standard deck

step 2

Determine the number of numbered cards in each suit

step 3

Calculate the total number of numbered cards in the deck

step 4

Compute the probability of drawing a numbered card from the deck

Answer

The probability of drawing a numbered card from a standard deck of 52 cards is

\frac{36}{52}

or

\frac{9}{13}

.

Key Concept

Probability of an Event

Explanation

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In a standard deck, there are 4 suits with numbered cards ranging from 2 to 10, which gives us 9 numbered cards per suit, totaling 36 numbered cards in the deck. Therefore, the probability of drawing a numbered card is the number of numbered cards (36) divided by the total number of cards (52).

5. Draw a Venn diagram to represent the relationship between the sets

X=\{1,2,5,6,7,9,10\}

and

Y=\{1,3

,

4,5,6,8,10\}

Solution by Steps

step 1

Identify the elements that are unique to set

X

and set

Y

, and those that are common to both sets

step 2

Set

X

unique elements:

\{2, 7, 9\}

; Set

Y

unique elements:

\{3, 4, 8\}

; Common elements:

\{1, 5, 6, 10\}

step 3

Draw two overlapping circles to represent sets

X

and

Y

step 4

Label the left circle as set

X

and the right circle as set

Y

step 5

In the left circle, outside the overlapping region, write the unique elements of set

X

:

2, 7, 9

step 6

In the right circle, outside the overlapping region, write the unique elements of set

Y

:

3, 4, 8

step 7

In the overlapping region, write the common elements:

1, 5, 6, 10

Answer

The Venn diagram is drawn with sets

X

and

Y

showing their unique and common elements.

Key Concept

Venn Diagram Representation

Explanation

A Venn diagram is used to visually represent the relationships between different sets, showing their common and unique elements.

If one card is chosen at random from a well-shuffled deck, what is the probability that the card is a king or a spade?

Solution by Steps

step 1

Identify the total number of kings and spades in a deck

step 2

There are 4 kings in a deck and 13 spades

step 3

Since one of the spades is a king, we must subtract this overlap to avoid double-counting

step 4

Calculate the probability by adding the number of favorable outcomes and dividing by the total number of outcomes

step 5

The probability is

\frac{4 + 13 - 1}{52}

step 6

Simplify the fraction to get the final probability

Answer

\frac{16}{52}

or

\frac{4}{13}

Key Concept

Inclusion-Exclusion Principle

Explanation

The probability of either event A or event B occurring is found by adding the probabilities of each event separately and then subtracting the probability of both events occurring together.

An experiment consists of drawing a number at random from

\{1,2,3,4,5,6,7,8\}

. Let

A=\{1,2,3,4\}, B=

\{1,3,5,7\}

and

C=\{4,6,8\}

. a. Are

A

and

B

independent? b. Are

\mathrm{A}

and

\mathrm{C}

independent? c. Are

B

and

C

independent?

Solution by Steps

step 1

Determine the probability of event

A

occurring

step 2

P(A) = \frac{\text{Number of outcomes in } A}{\text{Total number of outcomes}} = \frac{4}{8} = \frac{1}{2}

step 3

Determine the probability of event

B

occurring

step 4

P(B) = \frac{\text{Number of outcomes in } B}{\text{Total number of outcomes}} = \frac{4}{8} = \frac{1}{2}

step 5

Determine the probability of both events

A

and

B

occurring together

step 6

P(A \cap B) = \frac{\text{Number of outcomes in both } A \text{ and } B}{\text{Total number of outcomes}} = \frac{2}{8} = \frac{1}{4}

step 7

Check if

P(A \cap B) = P(A) \cdot P(B)

to determine if

A

and

B

are independent

step 8

Since

P(A \cap B) = \frac{1}{4}

and

P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

, the events

A

and

B

are independent

Answer

Yes, events

A

and

B

are independent.

Key Concept

Independence of Events

Explanation

Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other, which is mathematically expressed as

P(A \cap B) = P(A) \cdot P(B)

.

Solution by Steps

step 1

Determine the probability of event

A

occurring

step 2

P(A) = \frac{4}{8} = \frac{1}{2}

. (This is the same as in the previous calculation.)

step 3

Determine the probability of event

C

occurring

step 4

P(C) = \frac{\text{Number of outcomes in } C}{\text{Total number of outcomes}} = \frac{3}{8}

step 5

Determine the probability of both events

A

and

C

occurring together

step 6

P(A \cap C) = \frac{\text{Number of outcomes in both } A \text{ and } C}{\text{Total number of outcomes}} = \frac{1}{8}

step 7

Check if

P(A \cap C) = P(A) \cdot P(C)

to determine if

A

and

C

are independent

step 8

Since

P(A \cap C) = \frac{1}{8}

and

P(A) \cdot P(C) = \frac{1}{2} \cdot \frac{3}{8} = \frac{3}{16}

, the events

A

and

C

are not independent

Answer

No, events

A

and

C

are not independent.

Key Concept

Independence of Events

Explanation

Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other, which is mathematically expressed as

P(A \cap B) = P(A) \cdot P(B)

. In this case, since

P(A \cap C) \neq P(A) \cdot P(C)

, events

A

and

C

are not independent.

Solution by Steps

step 1

Determine the probability of event

B

occurring

step 2

P(B) = \frac{4}{8} = \frac{1}{2}

. (This is the same as in the first calculation.)

step 3

Determine the probability of event

C

occurring

step 4

P(C) = \frac{3}{8}

. (This is the same as in the second calculation.)

step 5

Determine the probability of both events

B

and

C

occurring together

step 6

P(B \cap C) = \frac{\text{Number of outcomes in both } B \text{ and } C}{\text{Total number of outcomes}} = \frac{0}{8} = 0

step 7

Check if

P(B \cap C) = P(B) \cdot P(C)

to determine if

B

and

C

are independent

step 8

Since

P(B \cap C) = 0

and

P(B) \cdot P(C) = \frac{1}{2} \cdot \frac{3}{8} = \frac{3}{16}

, the events

B

and

C

are not independent

Answer

No, events

B

and

C

are not independent.

Key Concept

Independence of Events

Explanation

Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other, which is mathematically expressed as

P(A \cap B) = P(A) \cdot P(B)

. In this case, since

P(B \cap C) \neq P(B) \cdot P(C)

, events

B

and

C

are not independent.

If

A

and

B

are independent events with

\operatorname{Pr}(A)=0.5

and

\operatorname{Pr}(B)=0.2

, find

\operatorname{Pr}(A \cup B)

Solution by Steps

step 1

Use the formula for the probability of the union of two events:

\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)

step 2

Since

A

and

B

are independent,

\Pr(A \cap B) = \Pr(A) \times \Pr(B)

step 3

Substitute the given probabilities into the formula:

\Pr(A \cup B) = 0.5 + 0.2 - (0.5 \times 0.2)

step 4

Calculate the intersection probability:

0.5 \times 0.2 = 0.1

step 5

Calculate the union probability:

0.5 + 0.2 - 0.1 = 0.6

Answer

\Pr(A \cup B) = 0.6

Key Concept

Probability of the Union of Two Independent Events

Explanation

For independent events

A

and

B

, the probability of their union is the sum of their individual probabilities minus the probability of their intersection. Since they are independent, the intersection probability is the product of their individual probabilities.

For the following diagram, find the marginal probability of

\mathrm{M}

\begin{tabular}{r|ccc|} &

H

&

N

&

L

\\ \hline

M

& 88 & 22 & 10 \\

F

& 11 & 22 & 12 \\ \hline Total & 99 & 44 & 22 \end{tabular}

Generated Graph

Solution by Steps

step 1

Calculate the total number of outcomes for

M

by adding the values in the

M

row

step 2

M_{total} = 88 + 22 + 10

step 3

M_{total} = 120

step 4

Calculate the sum of all outcomes in the table to find the grand total

step 5

Grand total

= 99 + 44 + 22

step 6

Grand total

= 165

step 7

Find the marginal probability of

M

by dividing the total number of outcomes for

M

by the grand total

step 8

Marginal probability of

M = \frac{M_{total}}{\text{Grand total}}

step 9

Marginal probability of

M = \frac{120}{165}

step 10

Simplify the fraction to get the marginal probability of

M

step 11

Marginal probability of

M = \frac{24}{33}

step 12

Marginal probability of

M = \frac{8}{11}

Answer

The marginal probability of

M

is

\frac{8}{11}

.

Key Concept

Marginal Probability

Explanation

The marginal probability of an event is the sum of the probabilities of all outcomes for that event divided by the total number of possible outcomes. In this case, it is the sum of the outcomes for

M

divided by the grand total of all outcomes.

Given that for two events

A

and

B, \operatorname{Pr}(A)=0.6, \operatorname{Pr}(B)=0.3

and

\operatorname{Pr}(B \mid A)=0.1

, find: a.

\operatorname{Pr}(A \cap B)

b.

\operatorname{Pr}(A \mid B)

Solution by Steps

step 1

To find

\operatorname{Pr}(A \cap B)

, we use the formula

\operatorname{Pr}(A \cap B) = \operatorname{Pr}(B \mid A) \cdot \operatorname{Pr}(A)

step 2

Substitute the given values into the formula:

\operatorname{Pr}(A \cap B) = 0.1 \cdot 0.6

step 3

Calculate the product to find

\operatorname{Pr}(A \cap B)

:

\operatorname{Pr}(A \cap B) = 0.06

Answer

\operatorname{Pr}(A \cap B) = 0.06

Key Concept

Multiplication Rule for Probability

Explanation

The probability of the intersection of two events is the product of the probability of one event and the conditional probability of the second event given the first.

Solution by Steps

step 1

To find

\operatorname{Pr}(A \mid B)

, we use the formula

\operatorname{Pr}(A \mid B) = \frac{\operatorname{Pr}(A \cap B)}{\operatorname{Pr}(B)}

step 2

Substitute the known values into the formula:

\operatorname{Pr}(A \mid B) = \frac{0.06}{0.3}

step 3

Calculate the division to find

\operatorname{Pr}(A \mid B)

:

\operatorname{Pr}(A \mid B) = 0.2

Answer

\operatorname{Pr}(A \mid B) = 0.2

Key Concept

Conditional Probability

Explanation

The conditional probability of an event A given event B is the probability of A and B occurring together divided by the probability of B.

In a particular school

55 \%

of the students are male and

45 \%

are female. Of the male students

13 \%

say mathematics is their favourite subject, while of the female students

18 \%

prefer mathematics. Find the probability that: a. a student chosen at random prefers mathematics and is female b. a student chosen at random prefers mathematics and is male

Solution by Steps

step 1

Calculate the probability that a student chosen at random is female and prefers mathematics

step 2

Use the given percentage of female students who prefer mathematics:

45\% \times 18\%

step 3

Convert percentages to decimals and multiply:

0.45 \times 0.18

step 4

According to asksia-ll calculation:

0.45 \times 0.18 = 0.081

step 5

The probability that a student chosen at random is female and prefers mathematics is

0.081

Answer

The probability that a student chosen at random prefers mathematics and is female is

0.081

.

Key Concept

Probability of combined events

Explanation

To find the probability of two combined events (being female and preferring mathematics), multiply the probabilities of each individual event.

Solution by Steps

step 1

Calculate the probability that a student chosen at random is male and prefers mathematics

step 2

Use the given percentage of male students who prefer mathematics:

55\% \times 13\%

step 3

Convert percentages to decimals and multiply:

0.55 \times 0.13

step 4

According to asksia-ll calculation:

0.55 \times 0.13 = 0.0715

step 5

The probability that a student chosen at random is male and prefers mathematics is

0.0715

Answer

The probability that a student chosen at random prefers mathematics and is male is

0.0715

.

Key Concept

Probability of combined events

Explanation

To find the probability of two combined events (being male and preferring mathematics), multiply the probabilities of each individual event.

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