MATHEMATICS HIGH SCHOOL

A deck of 52 cards contains four aces. If the cards are shuffled and distributed in a random manner to four players so that each player receives 13 cards, what is the probability that all four aces will be received by the same player?

Answers

Answer 1

Answer:

Answer:

The probability that all four aces will be received by the same player is approximately 0.01.

Step-by-step explanation:

Of the 52 cards the four aces can be selected in $52\choose4$ ways.

If any one of the players receives all the four aces then that player can be selected in $4\choose1$ ways.

Now for the selected player to receive all the four aces, the four ace cards must be placed among the 13 cards the player receives. This can be done in $13\choose4$ ways.

Then the total number of ways such that all the four aces is received by one player is $4$ $13\choose4$ .

Then the probability that all four aces will be received by the same player is:

$4*\frac{{13\choose4}}{{52\choose4}}=4*(715)/(270725) =0.01056\approx 0.01$

Thus, the probability that all four aces will be received by the same player is approximately 0.01.

Answer 2

Answer:

Final answer:

The probability that all four aces will be received by the same player in a game of 52 card deck distributed among four players is given by the formula P(E) = 4 * C(48, 9) / C(52, 13), derived through principles of combinatorics.

Explanation:

This is a probability problem that can be solved using combination and permutation principles in mathematics. Specifically, the subject involves combinatorics.

First, we note that each of the four players is dealt 13 cards from a 52 card deck. Hence, for a single player, the total number of ways 13 cards can be chosen from 52 is given by the combination formula C(n, r) = n! / [(n-r)!r!], where ! denotes factorial. In our case, n=52, and r=13.

The total number of ways to deal these 13 cards is thus C(52, 13).

Next, we need to consider the specific case where all four aces end up with one player. This means this player has 9 other cards that are not aces, from the remaining 48 cards (52 total cards - 4 aces). This can happen in C(48, 9) ways.

Therefore, the probability that a specific player gets all four aces is P(E) = C(48, 9) / C(52, 13).

To find the probability that any of the four players gets all four aces, we multiply this result by 4, because there are four players.

So, the final probability is P(E) = 4 * C(48, 9) / C(52, 13).

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Select the term that best describes the statement.If a polygon has eight sides, then it is an octagon.

conjunction
disjunction
negation
conditional

Answers

Answer:

The term that best describes the given statement is given by the last option i.e.

Conditional .

Step-by-step explanation:

We are given a statement:

If a polygon has eight sides, then it is an octagon.

We know that a conditional statement which is also known as the if-then statement is a statement where one part of it is a given condition or hypothesis and the second statement is a conclusion of the first statement.

It could also be symbolized as:

If p then q

where p is the hypothesis and q is the conclusion.

Hence, the term that describes this given statement is:

Conditional.

conditional
if the polygon has 6 sides then it is a hexagon, is an example of condition sencece so the correct answer is conditional.

Graph y = 5x and y = log5x on a sheet of paper using the same set of axes. Use the graph to describe the domain and range of each function. Then identify the y-intercept of each function and any asymptotes of each function. Explain also.

Answers

Answer:

1) For $y=5x$

A) Domain= $(-\infty,\infty) [ \left.\begin{matrix}x\end{matrix}\right|x\varepsilon \mathbb{R}]$

B) Range= $(-\infty,\infty) [ \left.\begin{matrix}y\end{matrix}\right|y\varepsilon \mathbb{R}]$

C) y-intercept = 0

D) Asymptote= No asymptote

2) For $y=log_5x$

A) Domain=Domain= $(0,\infty) [ \left.\begin{matrix}x\end{matrix}\right|x>0]$

B) Range= $(-\infty,\infty) [ \left.\begin{matrix}y\end{matrix}\right|y\varepsilon \mathbb{R}]$

C) y-intercept = None

D) Vertical Asymptote: x=0

Step-by-step explanation:

Given : $y=5x$ and $y=log_5x$

Refer the graph attached.

1) In equation (1) $y=5x$

→The domain is the set of all possible values in which function is defined.

y=5x is a linear polynomial defined on all real numbers.

Domain= $(-\infty,\infty) [ \left.\begin{matrix}x\end{matrix}\right|x\varepsilon \mathbb{R}]$

→Range is the set of all values that function takes.

It also include all real numbers.

Range= $(-\infty,\infty) [ \left.\begin{matrix}y\end{matrix}\right|y\varepsilon \mathbb{R}]$

→y-intercept- Value of y at the point where the line crosses the y axis.

put x=0 in equation y=5x we get, y=0

Therefore, y-intercept = 0 (We can see in the graph also)

→An asymptote is a line that a curve approaches, but never touches.

Asymptote= No asymptote

2) Now in equation (2) $y=log_5x$

Domain= $(0,\infty) [ \left.\begin{matrix}x\end{matrix}\right|x>0]$

because log function is not defined in negative.

Range= $(-\infty,\infty) [ \left.\begin{matrix}y\end{matrix}\right|y\varepsilon \mathbb{R}]$

y-intercept - None

Vertical Asymptote: x=0

A) Domain= (-∞, ∞) for all x

B) Range= (-∞, ∞) for all y

C) y-intercept = 0

D) Asymptote= No asymptote

A) Domain=(0, ∞) for all x > 0

B) Range= (-∞, ∞) for all y

C) y-intercept = None

D) Vertical Asymptote: x=0

Here, we have,

Function 1: y = 5x

Domain: The domain of this function is all real numbers because there are no restrictions on the values that x can take.

Range: The range of this function is also all real numbers because for every value of x, we can find a corresponding y value by multiplying it by 5.

Y-intercept: To find the y-intercept, we set x = 0 and solve for y. Substituting x = 0 into the equation, we get y = 5(0) = 0. Therefore, the y-intercept is (0, 0).

Asymptotes: There are no asymptotes in this linear function.

Function 2: y = log₅x

Domain: The domain of this function is all positive real numbers because the logarithm function is only defined for positive values of x.

Range: The range of this function is all real numbers because the logarithm function can produce any real number output.

Y-intercept: To find the y-intercept, we set x = 1 and solve for y. Substituting x = 1 into the equation, we get y = log₅(1) = 0. Therefore, the y-intercept is (0, 0).

Asymptotes: The logarithmic function has a vertical asymptote at x = 0 because the logarithm is undefined for x ≤ 0. Additionally, there is no horizontal asymptote.

When plotting these functions on the same set of axes, we will observe that the graph of y = 5x is a straight line passing through the origin (0, 0) with a slope of 5.

The graph of y = log₅x will appear as a curve that starts at the point (1, 0) and approaches the vertical asymptote x = 0 as x approaches zero.

The two graphs will intersect at the point (1, 0) because log₅1 = 0.

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What is the completely factored form of x^3 – 64x?a) x(x – 8)(x – 8)
b) (x - 4) ( x^2 + 4x + 16)
c) x(x – 8)(x + 8)
d) (x – 4)(x + 4)(x + 4)

Answers

The answer is option C.

Write 0.87 as a fraction in simplest form.

Answers

Write the decimal number as a fraction
(over 1)
0.87 = 0.87 / 1

Multiplying by 1 to eliminate 2 decimal places
we multiply top and bottom by 2 10's

Numerator (N)
N = 0.87 × 10 × 10 = 87
Denominator (D)
D = 1 × 10 × 10 = 100

N / D = 87 / 100

Simplifying our fraction

= 87/100

= 87/100

Curt just drank a cup of coffee to help him stay awake. The coffee had 130 milligrams of caffeine in it. If his body processes 7% of the caffeine every hour, how much will be left in 4 hours?

Answers

Answer: 97.25 milligrams

Step-by-step explanation:

Given: Curt just drank a cup of coffee to help him stay awake.

The initial amour of caffeine in coffee= 130 milligrams

The rate of decay of caffeine in body = 7%= 0.07

We know that the exponential decay equation with rate r and and time t is given by :-

$f(x)=A(1-r)^t$ , where A is the initial value.

Now, the amount of caffeine left in body after 4 hours is given by :-

$f(4)=130(1-0.07)^4\n\n\Rightarrow\ f(4)=97.2467613\approx97.25\text{ milligrams}$

Hence, after 4 hours 97.25 milligrams will be left.

After 4 hours, his body will have processed 28% because 7*4 = 28. But, you're trying to find what's left (so, subtract that % from 100)

100%-28% = 72%

Then multiply the percent left by the total number of milligrams of caffeine

130*.72 = 93.6 mg

A small swimming pool has a circumference of 3π feet. Select the true statement about the pool.a The radius is 3 ft.

b The diameter is 3 ft.

c The diameter is 1.5 ft.

d The radius is 6 ft.

Answers

Answer:

d dfghjkl

Step-by-step explanation:

i just knowvghnm,.

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