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Chapter 3: Q. 3.16 (page 107)

Independent trials that result in a success with probability pand a failure with probability 1-pare called Bernoulli trials. Let Pndenote the probability that n Bernoulli trials result in an even number of successes (0 being considered an even number). Show that

Pn=p1-Pn-1+(1-p)Pn-1n鈮�1

Short Answer

Expert verified

Condition upon the outcome of the first trial. Independence make the following trials, n-1Bernoulli trials.

For the step of the induction substitute explicit formula for Pninto recursive formula.

Step by step solution

01

conditions for the problems

Conditioning to the first trial can solve the problem.

If and only if one of the two disjoint situations below is true, we will get an even number of successes:

  1. The first trial was a success, and in n-1trials, an odd number of successes occurs.
  2. The first trial ended in failure, and n-1trials result in an even number of successes.
02

Independence between the trials

Because of the recursive relationship, the trials are completely independent of one another. in other words,

Pn=p1-Pn-1+qPn-1,鈭赌n鈮�1

n=1is included since we anticipated 0to be an even integer, hence P0=1.

03

Derivation by Induction

In the other hand, the demonstration by induction goes like this,

n=1,1+1-2p2=q=P1

n, For a universal natural number n, we suppose that the equality holds.

n+1, SubstitutingPn=1+(1-2p)n2

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Most popular questions from this chapter

Let S = {1, 2, . . . , n} and suppose that A and B are, independently, equally likely to be any of the 2n subsets (including the null set and S itself) of S.

(a) Show that

P{A 鈯�B} =34n

Hint: Let N(B) denote the number of elements in B. Use

P{A 鈯�B} =鈭�i=0nP{A (鈯�B|N(B) = i}P{N(B) = i}

Show that P{AB = 脴} =34n

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(a) You ask your friend if one of the cards is the ace of spades, and your friend answers in the affirmative.

(b) You ask your friend if the first card selected is an ace, and your friend answers in the affirmative.

(c) You ask your friend if the second card selected is an ace, and your friend answers in the affirmative.

(d) You ask your friend if either of the cards selected is an ace, and your friend answers in the affirmative.

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(a) What is the probability at least one red ball is chosen?

(b) Given that no red balls are chosen, what is the conditional probability that there are exactly 2 green balls among the 6 chosen?

The color of a person鈥檚 eyes is determined by a single pair of genes. If they are both blue-eyed genes, then the person will have blue eyes; if they are both brown-eyed genes, then the person will have brown eyes; and if one of them is a blue-eyed gene and the other a brown-eyed gene, then the person will have brown eyes. (Because of the latter fact, we say that the brown-eyed gene is dominant over the blue-eyed one.) A newborn child independently receives one eye gene from each of its parents, and the gene it receives from a parent is equally likely to be either of the two eye genes of that parent. Suppose that Smith and both of his parents have brown eyes, but Smith鈥檚 sister has blue eyes.

(a) What is the probability that Smith possesses a blue eyed gene?

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(c) If their first child has brown eyes, what is the probability that their next child will also have brown eyes?
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