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Chapter 3: Problem 80

A coin that comes up heads with probability \(p\) is flipped \(n\) consecutive times. What is the probability that starting with the first flip there are always more heads than tails that have appeared?

Short Answer

Expert verified
The probability of having more heads than tails at any point in the sequence of n flips (even n only) is given by: \[P(\text{more heads than tails}) = \frac{C_{n/2} \cdot p^{n/2 + 1} \cdot (1-p)^{n/2 - 1}}{2^n}\] Where \(C_{n/2}\) is the Catalan number for \(\frac{n}{2}\) and p is the probability of getting a head. For any odd n, the probability will be 0.

Step by step solution

01

Determine all possible outcomes

The first step is to determine the total number of possible outcomes for flipping a coin n times. Since there are 2 possible outcomes (heads and tails) for each flip, there are a total of 2^n possible outcomes.
02

Valid sequences

Now, we need to determine the number of valid sequences where we have more heads than tails at any given point. A valid sequence can only start with a head. For a sequence of length n, we have the following number of valid sequences: - If n is odd, there cannot be a valid outcome since there will always be an even number of tails. - If n is even, we can calculate the valid sequences using the Catalan numbers which are given by the formula \(C_{k} = \frac{1}{k+1} \binom{2k}{k}\), where k represents the position in the sequence of the Catalan numbers. In our case, k = n/2.
03

Calculate probability

To find the probability, we can divide the number of valid sequences by the total number of outcomes. Since we're given that the probability of getting a head on any given flip is p, the probability of getting a tail is (1-p). For the number of valid sequences, we use the Catalan number with k = n/2, as derived in the previous step. So, the probability of having more heads than tails at any point in the sequence is: \[P(\text{more heads than tails}) = \frac{C_{n/2} \cdot p^{n/2 + 1} \cdot (1-p)^{n/2 - 1}}{2^n}\] Keep in mind that the formula is valid only for even n, for any odd n the probability will be 0.

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Most popular questions from this chapter

Consider a sequence of independent trials, each of which is equally likely to result in any of the outcomes \(0,1, \ldots, m\). Say that a round begins with the first trial, and that a new round begins each time outcome 0 occurs. Let \(N\) denote the number of trials that it takes until all of the outcomes \(1, \ldots, m-1\) have occurred in the same round. Also, let \(T_{j}\) denote the number of trials that it takes until \(j\) distinct outcomes have occurred, and let \(I_{j}\) denote the \(j\) th distinct outcome to occur. (Therefore, outcome \(I_{j}\) first occurs at trial \(T_{j} .\) ) (a) Argue that the random vectors \(\left(I_{1}, \ldots, I_{m}\right)\) and \(\left(T_{1}, \ldots, T_{m}\right)\) are independent. (b) Define \(X\) by letting \(X=j\) if outcome 0 is the \(j\) th distinct outcome to occur. (Thus, \(I_{X}=0 .\) ) Derive an equation for \(E[N]\) in terms of \(E\left[T_{j}\right]\), \(j=1, \ldots, m-1\) by conditioning on \(X\) (c) Determine \(E\left[T_{j}\right], j=1, \ldots, m-1\). Hint: See Exercise 42 of Chapter 2 . (d) Find \(E[N]\).

\(A\) and \(B\) roll a pair of dice in turn, with \(A\) rolling first. A's objective is to obtain a sum of 6, and \(B\) 's is to obtain a sum of 7 . The game ends when either player reaches his or her objective, and that player is declared the winner. (a) Find the probability that \(A\) is the winner. (b) Find the expected number of rolls of the dice. (c) Find the variance of the number of rolls of the dice.

A coin having probability \(p\) of coming up heads is continually flipped. Let \(P_{j}(n)\) denote the probability that a run of \(j\) successive heads occurs within the first \(n\) flips. (a) Argue that $$ P_{j}(n)=P_{j}(n-1)+p^{j}(1-p)\left[1-P_{j}(n-j-1)\right] $$ (b) By conditioning on the first non-head to appear, derive another equation relating \(P_{j}(n)\) to the quantities \(P_{j}(n-k), k=1, \ldots, j\)

Suppose \(p(x, y, z)\), the joint probability mass function of the random variables \(X, Y\), and \(Z\), is given by $$ \begin{array}{ll} p(1,1,1)=\frac{1}{8}, & p(2,1,1)=\frac{1}{4}, \\ p(1,1,2)=\frac{1}{8}, & p(2,1,2)=\frac{3}{16}, \\ p(1,2,1)=\frac{1}{16}, & p(2,2,1)=0, \\ p(1,2,2)=0, & p(2,2,2)=\frac{1}{4} \end{array} $$ What is \(E[X \mid Y=2] ?\) What is \(E[X \mid Y=2, Z=1]\) ?

Suppose that \(X\) and \(Y\) are independent random variables with probability density functions \(f_{X}\) and \(f_{Y}\). Determine a one-dimensional integral expression for \(P\\{X+Y
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