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Chapter 15: Q7P (page 743)

What is the probability that the 2 and 3 of clubs are next to each other in a shuffled deck? Hint: Imagine the two cards accidentally stuck together and shuffled as one card.

Short Answer

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Answer

The probability that the 2 and 3 of clubs are next to each other in a shuffled deck is 126

Step by step solution

01

Given Information

A deck of 52 cards is given.

02

Definition of Independent Event

When the order of arrangement is definite, the permutation is applied and when the order is not definite,combination is applied.

03

Finding the probability that the 2 and 3 of clubs are next to each other in a shuffled deck

Let the 2 and 3 of club stick together and is assumed to become a single card, this implies that the deck will have 51 cards. And the cards can be arranged inand the stuck cards can be arranged in.

The total arrangements possible is.

This implies that the number of outcomes favourable areand total number of outcomes are.

Apply the formula for probability, that isto get the probability that the 2 and 3 of clubs are next to each other in a shuffled deck.

Thus the required probability is.

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Most popular questions from this chapter

Define s by the equation.s2=(1/n)∑i=1n(xi−x¯)2Show that the expected valueof.s2is[(n−1)/n]σ2Hints: Write

(xi−x¯)2=[(xi−μ)−(x¯−μ)]2=(xi−μ)2−2(xi−μ)(x¯−μ)+(x¯−μ)2

Find the average value of the first term from the definition ofσ2and the average value of the third term from Problem 2. To find the average value of the middle term write

(x¯−μ)=(x1+x2+⋯+xnn−μ)=1n[(x1−μ)+(x2−μ)+⋯+(xn−μ)]

Show by Problemthat

E[(xi−μ)(xj−μ)]=E(xi−μ)E(xj−μ)=0   fori≠j

andevaluate6.14 (same as the first term). Collect terms to find

E(s2)=n−1nσ2

By expandingw(x,y,z) in a three-variable power series similarto ,(10.10)show that

rw=(∂w∂x)2rx2+(∂w∂y)2ry2+(∂w∂z)2rz2

(a) One box contains one die and another box contains two dice. You select a box at random and take out and toss whatever is in it (that is, toss both dice if you have picked box 2 ). Let x=number of 3'sshowing. Set up the sample space and associated probabilities for x .

(b) What is the probability of at least one3?

(c) If at least one 3 turns up, what is the probability that you picked the first box?

(d) Find xand.σ

(a) Suppose you have two quarters and a dime in your left pocket and two dimes and three quarters in your right pocket. You select a pocket at random and from it a coin at random. What is the probability that it is a dime? (b) Let x be the amount of money you select. Find E(x).

(c) Suppose you selected a dime in (a). What is the probability that it came from your right pocket?

(d) Suppose you do not replace the dime, but select another coin which is also a dime. What is the probability that this second coin came from your right pocket?

(a) Find the probability density function f(x)for the position x of a particle which is executing simple harmonic motion on (−a,a)along the x axis. (See Chapter 7 , Section 2 , for a discussion of simple harmonic motion.) Hint: The value of x at time t is x=acosωt. Find the velocity dxdt ; then the probability of finding the particle in a given dx is proportional to the time it spends there which is inversely proportional to its speed there. Don’t forget that the total probability of finding the particle somewhere must be 1.

(b) Sketch the probability density function f(x)found in part (a) and also the cumulative distribution function f(x) [see equation (6.4)].

(c) Find the average and the standard deviation of x in part (a).
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