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Chapter 15: Q5P (page 724)

Given a family of two children (assume boys and girls equally likely, that is, probability for each), what is the probability 1/2 that both are boys? That at least one is a girl? Given that at least one is a girl, what is the probability that both are girls? Given that the first two are girls, what is the probability that an expected third child will be a boy?

Short Answer

Expert verified

The probability that both are boys is 1/4.

The probability that at least one is a girl is 3/4.

When at least one is a girl, the probability that both are girls is 1/3.

When the first two are girls, the probability that an expected third child will be a boy is 1/2.

Step by step solution

01

Expression for the Probability

The probability for an event E is expressed as follows,

P = number of outcomes favorable to E/total number of outcomes ...(i)

02

Determination of the probability that both are boys

A child being a girl or a boy does not depend on whether the previous child was a girl or a boy.

Let G represents girl child and B represents Boy child, thus the possible outcomes for two children are 4 namely (BB, BG, GB, GG).

In possible outcome, only one case namely (BB) is the favorable outcome, this implies that the probability that both are boys is 1/4.

03

Determination of the probability that at least one is a girl 

In possible outcome, only three cases namely (GG, GB, BG) are the favorable outcome, this implies that the probability that at least one is a girl is 3/4.

04

Determination of the probability that both are girls when at least one is a girl

In possible outcomes, only three cases namely (GG, GB, BG) has the at least one girl, out of which only one case namely (GG) is the favorable outcome, this implies that the probability that both are girls when at least one is a girl is 1/3.

05

Determination of the probability that an expected third child will be a boy given that the first two children are girls 

The probability of the third child being a boy is 1/2 as it does not depend whether the previous children were boy or girl.

Thus, the probability that both are boys is 1/4, the probability that at least one is a girl is 3/4, the probability that both are girls when at least one is a girl is 1/3, and the probability of the third child being a boy is 1/2.

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Most popular questions from this chapter

:(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鈥檚 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.

Two cards are drawn at random from a shuffled deck.

  1. What is the probability that at least one is a heart?

(b) If you know that at least one is a heart, what is the probability that both are

hearts?

A thick coin has probability 37of falling heads, 37of falling tails, and 17of standing one edge. Show that if it is tossed repeatedly it has probability 1 of

Eventually standing on edge.

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鈥檚 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.

How that the expectation of the sum of two random variables defined over the same sample space is the sum of the expectations. Hint: Let p1,p2,....,pnbe the probabilities associated with the n sample points; letx1,x2,....,xn and y1,y2,....,yn, be the values of the random variables x and y for the n sample points. Write out E(x),E(y),E(x+y).
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