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Chapter 3: Problem 7

The king comes from a family of 2 children. What is the probability that the other child is his sister??????? probability that the other child his? ??????? the ??

Short Answer

Expert verified
The probability that the other child is the king's sister is \(\frac{2}{3}\) or approximately 0.67.

Step by step solution

01

List the possible combinations of genders for 2 children

To do this, we will create a sample space and enumerate all the possible combinations of genders: 1. Male, Male (MM) 2. Male, Female (MF) 3. Female, Male (FM)
02

Identify the combinations with the king being a male

As the king is male, we focus on the combinations with at least one male child: 1. Male, Male (MM) 2. Male, Female (MF) 3. Female, Male (FM)
03

Determine the number of combinations that have a female sibling for the king

In the sample space, we will identify the possibilities where the other child is the king's sister (female): 1. Male, Female (MF) 2. Female, Male (FM)
04

Identify the total number of possible combinations and the combinations with a sister

There are a total of 3 possible combinations of genders and 2 combinations that include a sister.
05

Calculate the probability of the other child being the king's sister

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. So, the probability that the other child is the king's sister is given by: Probability = (Number of combinations with a sister) / (Total number of possible combinations) = \(\frac{2}{3}\) Therefore, the probability that the other child is the king's sister is \(\frac{2}{3}\) or approximately 0.67.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Sample Space
The concept of sample space is fundamental in probability. It represents the set of all feasible outcomes for a given experiment or scenario. In our exercise, we are examining a family with two children. Hence, we need to list all potential gender combinations for these two children. Here, we have the following scenarios:
  • Male, Male (MM)
  • Male, Female (MF)
  • Female, Male (FM)
In probability, the sample space is crucial as it is the foundation from which probabilities are calculated. Each combination is considered a possible outcome. Thus, understanding the complete sample space enables us to identify specific outcomes of interest later in our calculation processes.
Exploring Combinations
Combinations are about selecting items from a larger set in a way where the order of items does not matter. In the context of this exercise, we're interested in the gender pairs of two children. Here, we identified three combinations: MM, MF, and FM. Each represents a distinct way to have two children with their respective genders.

It's important to note in this example that MF and FM are considered different because they represent different birth orders. While combinations help in organizing possible outcomes, they highlight how certain constraints鈥攕uch as birth order or gender鈥攃an affect our final probability calculations.

Using combinations efficiently involves careful enumeration of possible outcomes and considering each unique variation. This ensures no potential situation is overlooked when assessing probabilities.
Favorable Outcomes in Probability
Favorable outcomes in probability refer to the specific outcomes that satisfy the condition we鈥檙e interested in. In this particular scenario, we are focused on the probability of the other child being the king's sister.

From our sample space, we identified that these outcomes are the ones where a female appears alongside the king. The gender combinations' favorable outcomes are:
  • Male, Female (MF)
  • Female, Male (FM)
These are the scenarios where the king, being male, has a sister.

Calculating probability involves the ratio of these favorable outcomes to the total number of outcomes in the sample space. In this case, there are 2 favorable outcomes over a total of 3 possible combinations. Consequently, the probability is computed as follows: \(\frac{2}{3}\). This represents a high likelihood, demonstrating that the probability reflects both the sample space and favored conditions.

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Most popular questions from this chapter

A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene, in the sense that if an organism possesses the gene pair \(x X\) then it will outwardly have the appearance of the \(X\) gene. For instance, if \(X\) stands for brown eyes and \(x\) for blue eyes, then an individual having either gene pair \(X X\) or \(x X\) will have brown eyes, whereas one having gene pair \(x x\) will have blue eyes. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus, 2 organisms with respective genotypes \(a A, b B, c c, d D\) ee and \(A A, B B, c c, D D,\) ee would have different genotypes but the same phenotype.) In a mating between 2 organisms, each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of the organism's mate. In a mating between organisms having genotypes \(a A, b B, c C, d D, e E\) and \(a a, b B, c c\) \(D d,\) ee what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble (a) the first parent? (b) the second parent? (c) either parent? (d) neither parent?

On rainy days, Joe is late to work with probability \(.3 ;\) on nonrainy days, he is late with probability . \(1 .\) With probability.7, it will rain tomorrow. (a) Find the probability that Joe is early tomorrow. (b) Given that Joe was early, what is the conditional probability that it rained?

Consider an unending sequence of independent trials, where each trial is equally likely to result in any of the outcomes \(1,2,\) or \(3 .\) Given that outcome 3 is the last of the three outcomes to occur, find the conditional probability that. (a) the first trial results in outcome 1. (b) the first two trials both result in outcome \(1 .\)

Genes relating to albinism are denoted by \(A\) and a. Only those people who receive the \(a\) gene from both parents will be albino. Persons having the gene pair \(A, a\) are normal in appearance and, because they can pass on the trait to their offspring, are called carriers. Suppose that a normal couple has two children, exactly one of whom is an albino. Suppose that the nonalbino child mates with a person who is known to be a carrier for albinism. (a) What is the probability that their first offspring is an albino? (b) What is the conditional probability that their second offspring is an albino given that their firstborn is not?

In a certain contest, the players are of equal skill and the probability is \(\frac{1}{2}\) that a specified one of the two contestants will be the victor. In a group of \(2^{n}\) players, the players are paired off against each other at random. The \(2^{n-1}\) winners are again paired off randomly, and so on, until a single winner remains. Consider two specified contestants, \(A\) and \(B\), and define the events \(A_{i}, i \leq n, E\) by \(A_{i}:\) \(A\) plays in exactly \(i\) contests: \(E: \quad A\) and \(B\) never play each other. (a) \(\operatorname{Find} P\left(A_{i}\right), i=1, \ldots, n\) (b) Find \(P(E)\) (c) Let \(P_{n}=P(E) .\) Show that $$ P_{n}=\frac{1}{2^{n}-1}+\frac{2^{n}-2}{2^{n}-1}\left(\frac{1}{2}\right)^{2} P_{n-1} $$ and use this formula to check the answer you obtained in part (b). Hint: Find \(P(E)\) by conditioning on which of the events \(A_{i}, i=1, \ldots, n\) occur. In simplifying your answer, use the algebraic identity $$ \sum_{i=1}^{n-1} i x^{i-1}=\frac{1-n x^{n-1}+(n-1) x^{n}}{(1-x)^{2}} $$ For another approach to solving this problem, note that there are a total of \(2^{n}-1\) games played. (d) Explain why \(2^{n}-1\) games are played. Number these games, and let \(B_{i}\) denote the event that \(A\) and \(B\) play each other in game \(i, i=1, \ldots, 2^{n}-1\) (e) What is \(P\left(\bar{B}_{i}\right) ?\) (f) Use part (e) to find \(P(E).\)
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