**Probability with Dice Rolls - Understanding an Example**In this exercise, we explore the probability of a specific event occurring when rolling a fair, 6-sided die four times in succession. The outcomes of these rolls are recorded in a list as $(a_1, a_2, a_3, a_4, a_5, a_6)$.**Problem Statement:**A fair 6-sided die is rolled four times in succession and the results are inserted into a list $(a_1, a_2, a_3, a_4, a_5, a_6)$. What is the probability that $a_4 \geq a_5 \geq a_6$?**Explanation:**In this problem, we are interested in finding the likelihood that the value of the fourth roll is greater than or equal to the fifth roll, and the fifth roll is greater than or equal to the sixth roll. Given that each roll is independent, we can leverage our understanding of permutations and combinations to solve this.**Step-by-Step Approach:**1. **Identify Outcomes:** Each of the rolls $a_1, a_2, a_3, a_4, a_5, a_6$ can take any integer value from 1 to 6.2. **Total Possible Combinations:** When rolling the die four times, there are a total of $6 \times 6 \times 6 \times 6 = 6^4$ possible outcomes for the sequence.3. **Desired Event:** The event of interest is $a_4 \geq a_5 \geq a_6$ among the recorded values.Given this setup, the solution involves calculating the proportion of outcomes that satisfy the condition $a_4 \geq a_5 \geq a_6$ out of all possible outcomes.This example enhances the understanding of probability concepts applied to dice, especially involving sequences and comparisons among rolled values. For a deeper dive into the specific calculations and enumerative combinatorics involved, further study into these fields of probability theory is recommended.

To calculate the probability that a4 ≥ a5 ≥a6 when rolling a fair 6-sided die four times, we need to…

Answered: A fair 6-sided die is rolled four times…

A fair 6-sided die is rolled four times in succession and the results are inserted into a list (a1, A2, A3, A4, A5, a6). What is the probability that a4 ≥ a5 ≥ a6?

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Counting And Probability

Section: Chapter Questions

Problem 2P: Family Planning A intend to have two children. What is tie probability that they will have child of...

See similar textbooks

Related questions

Q: Set up a system of linear equations in two variables that models the problem. Then solve the system…

A: A system of linear equations refers to a set of equations that involve multiple variables, and the…

Q: ertain pickup truck by GMC comes with six body styles, three drive types, eight exterior colors, two…

A: Given :A certain pickup truck by GMC comes with six body styles, three drive types, eight exterior…

Q: (Consider the function f: R R satisfying [x² - 4x +4, ifx ≥ 2 La(x), if x < 2 Define a (x) such that…

Q: Let u = 6 - 1 and v= 6 Show that a a -[:] k Let b = k How can it be shown that a vector b is in Span…

Q: A) 4 2 (9) The derivative of 2sin-¹x (2sin-1x) In 2 A) √1-x2 B) is: 2sin 1x √1-x² (2sin 1x) In 2 C)…

A: The given problem is to find the derivative of the given function.Given function,2 sin^(-1)x

Q: 3.3 2 3 Given: 5 (¹) + 10 (¹)² + 20 (1-4) 0 (¹-4)³ + + ... 3.3.1 For which values of k will the…

Q: The set S = {(2a, b, a - b): a, b = R} quos & jonai (o) tails wol2 woled old is a subspace of R³.…

Q: Solve the following system of equations using the Gaussian algorithm with column pivoting: a + 2b…

A: Let us first reduce the augmented matrix of given system into echelon form.

Q: Suppose a movie ticket cost $11 in 2019. What was its price in 1984 dollars? Assume that all prices…

A: According to the given data, find the movie ticket price in 1984.

Q: Given a function f (x)=e2 x which is a solution of following differential equation…

A: Differential equations

Q: 2. Express V and V.V in spherical coordinates.

A: Expression of gradient and divergent operator:

Q: Theorem 2.1 For every graph G, o (G) ≤ x (G). Proof It suffices to assume that G is a nontrivial…

Q: Let f(x) be a piecewise function of period 2L, f(x) -I, -L<x<0 0<x<L 2x, and let its corresponding…

Q: 2+4n 8 Let {n} be a sequence such that (1 + 2 ) ≤ an ≤ e ²2n n=1 following statements is correct?…

A: The sandwich theorem, or squeeze theorem, for real sequences is the statement that if (an) , (bn) ,…

Q: 4" (18) The coefficient of x³ in the Maclaurin series expansion of exis: A) - B) D) 3 (19) The…

Q: For the function f'(2) for f(x) = 3x + 1 2x - 3 approximate f'(a) in the following ways. (Round your…

A: The given function .To find the value of using the given methods.

Q: Compute the derivative (r₁(1) r2(1)) if r₁(t) = (t¹0, 1³, tº) 73 10t r2(t) = (et, e³t, elor) (Use…

Q: The graph of y = f(x) is graphed below. What is the end behavior of f(x)? y 236.6 207.02 177.45…

A: A graph of the function is given.We have to find the end behavior of the given function.

Q: Task 3 a. Simplify the following equation: G=1xe/²x2xe/0.5 x0.5× e/0.75 b. Express the following…

A: Given that From Euler's formula:…

Q: Which of the following functions is not even? f(x)=√x² - 1 1 f(x) = ₁ 1 + x² f(x) = cosecx f(x) =…

A: Functions :

Q: Use a truth table to determine whether the argument is valid or invalid. -p1q -q-p q^~P

A: In the truth table if there is no ROW in which conclusion is FALSE when both premises are true then…

Q: y' −²+²²2² = y X X = 0

A: The given Bernoulli equation is…

Q: Find the general solution of x² (y' + y2) + xy + x²-1=0 given that Y₁ = -tan(x) is a solution of…

Q: 20% of the eggs at a stall are large, 25% of the remaining ones are medium and the rest are small.…

A: Given that20% of the eggs at a stall are large , 25 % of the remaining ones are medium and the rest…

Q: 1.3. For this exercise, use the simple substitution table given in Table 1.11. (a) Encrypt the…

A: This question is related to the field of cryptography.

Q: Consider the statements below. Verify which statements are true and which ones are false. Justify…

Q: BOOK CLUB Four women—Teddi, Sherwood, Luann, and Kathi—have all been in a book club for several…

A: According to the question, Let's examine each clue one by oneTeddi liked The Poisonwood Bible, but…

Q: what is an example of: i) an edge e of a 2-connected graph G such that G/e is not 2-connected ii)…

A: i) An example of an edge e of a 2-connected graph G such that G/e is not 2-connected can be…

Q: Find an equation of the tangent line to y = f(x) at x = -2. (Using the definition) f(x)=x² + x² + x…

A: we have, Differentiating w.r.t x, we getDifferentiating again w.r.t x, we getFurther, . .…

Q: dT Newton's law of cooling states that the rate of change in the temperature T(t) of a body is…

Q: (22) The radius of convergence of A) 2√3 B) √3 (22) Resu 100 (x-4)2k C) 3 D) NIS D) x

A: (.) Given ,radius of convergence of power series .(.) Radius of convergence: The radius of…

Q: ΣΑ x n=1 (x-2) 2) n 3" (n² +1 2

Q: Let r(t) = (sin(t), cos(t), 6 sin(t) + 9 cos(2t)). Find the projection of r(t) onto the xz-plane…

A: Given vector…

Q: (5) Let v = 2 sine. a) Find where the curve horizontal tangent lines. b) Find where the vertical…

Q: 2. Calculate fxyzdz, where L is the circumference of the sphere x² + y² + z² = 1 truncated by the…

A: As per the question, we are asked to calculate the line integral of the function xyz along a…

Q: 5. Convert the function f(x)=eux and f(x+2) = f(x) defined in the interval (-π < x < π) to the…

A: The given problem is to find the complex Fourier series for the given periodic function.Given…

Q: 1.calculate ſ₁ (x² − y) dx + (y² − x)dy, where L is the left semicircle with (0, 2R) as the centre…

Q: (106) (i) Find the image of and (ii) the preimage of under the linear transformation with matrix 0…

A: The matrix of the given linear matrix is:We need to find the image of the and the preimage of the…

Q: Sketch several level curves of the function f(x, y) = xy. Use k = -4,-1, 0, 1, 4.

A: Function is .To Find:Plot the graph of the level curve of the given function for k=-4, -1, 0, 1, 4.

Q: the problem given is TT i=0 to n (1/(2i+1)*(1/2i+2) = 1(2n+2)!

Q: Evaluate the limit for r(t) = (-8, sin(t), 8). r(t + h) - r(t) h lim h→0 (Use symbolic notation and…

Q: S₁- X 5. Let S be the capped cylindrical surface shown in Figure 8.2.14. S is the union of two…

A: Stokes theorem:

Q: What is the Hip Law of Cosines Angle X Y 2.13 0.6 2.03 0.75 2.16 1.36 1.77 1.93 1.90 2.74 Metatarsal…

A: Given Data:To Find: Hip Angle

Q: Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points…

Q: Find the surface area of the prism. Round to the nearest tenth if necessary.

A: To find the surface area of a prism, you need to know the number of faces and their dimensions. A…

Q: Suppose that (a)=1 is a sequence and n Σa converges to 12. Let sn= Σak. n=1 k = 1 Which of the…

A: Sequence converges to 12 and Let To Find:Which of the following option is correct:a) b) Divergence…

Q: Classify each system as having zero, one or infinitely many solutions. ₁:4x-6y=-10 [x, y] =…

Q: Let r = (e²t-8, e4-6t, (t+1)-6). Compute the derivative. (Use symbolic notation and fractions where…

Q: Use hyperbolic functions to parametrize the intersection of the surfaces x² - y² = 4 and z = 5xy.…

A: Use hyperbolic functions to parametrize the intersection of the surfaces and (Use symbolic notation…

Q: find the principal curvatures and principal vectors of the saddle surface M:z = xy at the origin.

A: To find the principal curvatures and principal vectors of the saddle surface M: z = xy at the…

Question

**Probability with Dice Rolls - Understanding an Example**

In this exercise, we explore the probability of a specific event occurring when rolling a fair, 6-sided die four times in succession. The outcomes of these rolls are recorded in a list as $(a_1, a_2, a_3, a_4, a_5, a_6)$.

**Problem Statement:**
A fair 6-sided die is rolled four times in succession and the results are inserted into a list $(a_1, a_2, a_3, a_4, a_5, a_6)$. What is the probability that $a_4 \geq a_5 \geq a_6$?

**Explanation:**

In this problem, we are interested in finding the likelihood that the value of the fourth roll is greater than or equal to the fifth roll, and the fifth roll is greater than or equal to the sixth roll. Given that each roll is independent, we can leverage our understanding of permutations and combinations to solve this.

**Step-by-Step Approach:**

1. **Identify Outcomes:** Each of the rolls $a_1, a_2, a_3, a_4, a_5, a_6$ can take any integer value from 1 to 6.
2. **Total Possible Combinations:** When rolling the die four times, there are a total of $6 \times 6 \times 6 \times 6 = 6^4$ possible outcomes for the sequence.
3. **Desired Event:** The event of interest is $a_4 \geq a_5 \geq a_6$ among the recorded values.

Given this setup, the solution involves calculating the proportion of outcomes that satisfy the condition $a_4 \geq a_5 \geq a_6$ out of all possible outcomes.

This example enhances the understanding of probability concepts applied to dice, especially involving sequences and comparisons among rolled values.

For a deeper dive into the specific calculations and enumerative combinatorics involved, further study into these fields of probability theory is recommended.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1: Introduction

VIEW

Step 2: Step-by-Step Explanation

VIEW

Solution

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

YOUR TURN 7 Find the probability that when two fair dice are rolled, the sum is less than 11.
Family Planning A intend to have two children. What is tie probability that they will have child of each sex? The French mathematician IY Alembert analyzed this problem (incorrectly) by reasoning that three outcomes are possible: or two girls, or one child of each sex. He concluded that probability of having of each sex is mistakenly assuming that three outcomes are likely. (a) Model this problem with a pair of coins (using "heads" for boys and "tails" for girls). or write a program to model of problem. perform the experiment 40 mere times. Counting the number of boy-girl combinations. Estimate the probability of having child of each. (b) Calculate the correct probability of having one child of each sex and compare this with your result from part (a).
Dividing a Jackpot A game between two pIayers consists of tossing coin. Player A gets a point if the coin shows heads, and player B gets a point if it shows tails. The first player to get six points wins an $8000 jackpot. As it happens, the police raid the place when player A has five points and B has three points. After everyone has calmed down, how should the jackpot be divided between the two players? In other words, what is the probability of A winning (and that of B winning) if the game were to continue? The French mathematicians Pascal and Fermat corresponded about this problem, and both came to the same correct conclusion (though by very different reasoning's). Their friend Roberval disagreed with both of them. He argued that player A has probability of Winning, because the game can end in the four ways H, TH, TTH, TTT, and in three of these, A wins. Roberval’s reasoning was wrong. Continue the game from the point at which it was interrupted, using either a coin or a modeling program. Perform this experiment 80 or more times, and estimate the probability that player A wins. Calculate the probability that player A wins. Compare with your estimate from part (a).
The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different?
In Example 8, what is the probability that an employee chosen at random has 30 or more years of service?