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Chapter 15: Q7MP (page 776)

Suppose a coin is tossed three times. Let x be a random variable whose value is 1 if the number of heads is divisible by 3, and 0 otherwise. Set up the sample space for x and the associated probabilities. Find x and σ.

Short Answer

Expert verified

The solution has been derived asσ=34

Step by step solution

01

Given Information.

It has been given that a coin is tossed three times.

02

Definition of Probability. 

Probability means the chances of any event to occur is called it probability.

03

Find the probability. 

The sample space is given as described below.

S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}

Assume x =the number of heads divisible by 3 so values of x are given by the expression mentioned below.

x={0,1}

The probability of getting number of heads divisible by 3.

p(x=1)=28   P(x=0)=68

The meanis given by the expression mentioned below.

μ=∑xp(x)=1×28=14

Write the standard deviation.

var(x)=σ2=∑(x−μ)2p(x)=2(1−14)2(18)+6(14)2(18)=316

Solve for standard deviation.

σ=var(x)=316=34

Hence, the solution has been derived as.σ=34

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Most popular questions from this chapter

Question: Use both the sample space (2.4) and the sample space (2.5) to answer the following questions about a toss of two dice.

(a) What is the probability that the sum is ≥ 4?

(b) What is the probability that the sum is even?

(c) What is the probability that the sum is divisible by 3?

(d) If the sum is odd, what is the probability that it is equal to 7?

(e) What is the probability that the product of the numbers on the two dice is 12?

:(a) Suppose that Martian dice are regular tetrahedra with vertices labelled 1 to 4. Two such dice are tossed and the sum of the numbers showing is even. Let x be this sum. Set up the sample space for x and the associated probabilities.

(b) Find E(x) and.

(c) Find the probability of exactly fifteen 2’s in 48 tosses of a Martian die using the binomial distribution.

(d) Approximate (c) using the normal distribution.

(e) Approximate (c) using the Poisson distribution.

Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use itto solve the problem. Use either a uniform or non-uniform sample space or try both.

A trick deck of cards is printed with the hearts and diamonds black, and the spadesand clubs red. A card is chosen at random from this deck (after it is shuffled). Findthe probability that it is either a red card or the queen of hearts. That it is eithera red face card or a club. That it is either a red ace or a diamond.

(a) A loaded die has probabilities121,221,321,421,521,621of showing 1, 2, 3, 4, 5, 6.What is the probability of throwing two 3’s in succession?

(b) What is the probability of throwing a 4 the first time and not a 4 the second

Time with a die loaded as in (a)?

(c) If two dice loaded as in (a) are thrown, and we know that the sum of the

numbers on the faces is greater than or equal to 10, what is the probability

That both are 5s?

(d) How many times must we throw a die loaded as in (a) to have probability greater than 12of getting an ace?

(e) Adie, loaded as in (a), is thrown twice. What is the probability that thenumber on the die is even the first timethe second time?

Given a non uniform sample space and the probabilities associated with the points, we defined the probability of an event A as the sum of the probabilities associated with the sample points favorable to A. [You used this definition in Problem 15with the sample space (2.5).] Show that this definition is consistent with the definition by equally likely cases if there is also a uniform sample space for the problem (as there was in Problem 15). Hint: Let the uniform sample space have n<Npoints each with the probability N-1. Let the nonuniform sample space have n points, the first point corresponding to N1 points of the uniform space, the second to N2 points, etc. What is N1 + N2 + .... Nn ?What are p1, p2, ...the probabilities associated with the first, second, etc., points of the nonuniform space? What is p1 + p2 +....+ pn? Now consider an event for which several points, say i, j, k, of the nonuniform sample space are favorable. Then using the nonuniform sample space, we have, by definition of the probability p of the event, p = pi + pj + pk . Write this in terms of the N’s and show that the result is the same as that obtained by equally likely cases using the uniform space. Refer to Problem 15as a specific example if you need to.
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