Problem 21
Question
A small community organization consists of 20 families, of which 4 have one child, 8 have two children, 5 have three children, 2 have four children, and 1 has five children. (a) If one of these families is chosen at random, what is the probability it has \(i\) children, \(i=\) \(1,2,3,4,5 ?\) (b) If one of the children is randomly chosen, what is the probability that child comes from a family having \(i\) children, \(i=1,2,3,4,5 ?\)
Step-by-Step Solution
Verified Answer
(a) The probabilities of a randomly chosen family having $i$ children are:
\(P(i=1) = \frac{1}{5}\), \(P(i=2) = \frac{2}{5}\), \(P(i=3) = \frac{1}{4}\), \(P(i=4) = \frac{1}{10}\), and \(P(i=5) = \frac{1}{20}\).
(b) The probabilities of a randomly chosen child belonging to a family with $i$ children are:
\(P(\text{child in family with }i=1\text{ child}) = \frac{1}{12}\), \(P(\text{child in family with }i=2\text{ children}) = \frac{1}{3}\), \(P(\text{child in family with }i=3\text{ children}) = \frac{5}{16}\), \(P(\text{child in family with }i=4\text{ children}) = \frac{1}{6}\), and \(P(\text{child in family with }i=5\text{ children}) = \frac{5}{48}\).
1Step 1: A. Probability of Random Family having i children#
To find the probability that a randomly chosen family has i children, we will divide the number of families having i children by the total number of families. We will do this for i = 1, 2, 3, 4, and 5.
For i = 1:
P(i=1) = Number of families with 1 child / Total number of families = 4/20 = 1/5
For i = 2:
P(i=2) = Number of families with 2 children / Total number of families = 8/20 = 2/5
For i = 3:
P(i=3) = Number of families with 3 children / Total number of families = 5/20 = 1/4
For i = 4:
P(i=4) = Number of families with 4 children / Total number of families = 2/20 = 1/10
For i = 5:
P(i=5) = Number of families with 5 children / Total number of families = 1/20
These probabilities represent the likelihood of a randomly chosen family having i children.
2Step 2: B. Probability of a Random Child belonging to a family with i children#
To find the probability that a randomly chosen child belongs to a family with i children, we will use conditional probability.
First, we must find the total number of children in the community. We will multiply the number of families having i children by i, and add the results for i = 1, 2, 3, 4, and 5:
Total children = (4 x 1) + (8 x 2) + (5 x 3) + (2 x 4) + (1 x 5) = 4 + 16 + 15 + 8 + 5 = 48
Now, we will find the probability that a randomly chosen child belongs to a family with i children by dividing the number of children in families with i children (i x number of families with i children) by the total number of children:
For i = 1:
P(child in family with 1 child) = Number of children in families with 1 child / Total number of children = (4 x 1) / 48 = 4/48 = 1/12
For i = 2:
P(child in family with 2 children) = Number of children in families with 2 children / Total number of children = (8 x 2) / 48 = 16/48 = 1/3
For i = 3:
P(child in family with 3 children) = Number of children in families with 3 children / Total number of children = (5 x 3) / 48 = 15/48 = 5/16
For i = 4:
P(child in family with 4 children) = Number of children in families with 4 children / Total number of children = (2 x 4) / 48 = 8/48 = 1/6
For i = 5:
P(child in family with 5 children) = Number of children in families with 5 children / Total number of children = (1 x 5) / 48 = 5/48
These probabilities represent the likelihood of a randomly chosen child belonging to a family with i children.
Key Concepts
Conditional ProbabilityProbability DistributionRandom Selection
Conditional Probability
Understanding conditional probability is crucial when studying complex events, where the outcome is influenced by the presence of other conditions. It deals with calculating the probability that an event A occurs given that another event B has already occurred.
Let's use the problem at hand as an illustration. When we ask: 'What is the probability that a randomly chosen child comes from a family with a certain number of children?', we are addressing a conditional probability question. We are not just looking for any child; we are looking for a child who comes from a specific family size, which is a condition that affects the probability.
In our exercise, the calculation involved first determining the total number of children across all families, which served as our sample space for the random selection of a child. Then we computed the probability for each family size by accounting for the number of children in such families, dividing this by the total number of children in the community. This approach embodies the essence of conditional probability, which is applying additional information (the family size) to refine our probability assessment.
Let's use the problem at hand as an illustration. When we ask: 'What is the probability that a randomly chosen child comes from a family with a certain number of children?', we are addressing a conditional probability question. We are not just looking for any child; we are looking for a child who comes from a specific family size, which is a condition that affects the probability.
In our exercise, the calculation involved first determining the total number of children across all families, which served as our sample space for the random selection of a child. Then we computed the probability for each family size by accounting for the number of children in such families, dividing this by the total number of children in the community. This approach embodies the essence of conditional probability, which is applying additional information (the family size) to refine our probability assessment.
Probability Distribution
A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. It is an essential concept in statistics as it provides a full picture of the likelihood of various outcomes. In the context of our exercise, the probability distribution is constructed for the number of children in randomly selected families.
The first part of the exercise effectively creates a probability distribution when it lists the probabilities for families having one, two, three, four, and five children. To do this, we divide the number of families with a given number of children by the total number of families. This yields a simple distribution, with probabilities reflecting the likelihood of encountering families of various sizes in the sample. For practical applications and deeper understanding, graphical representations such as histograms or bar charts are often utilized to visualize the distribution.
The first part of the exercise effectively creates a probability distribution when it lists the probabilities for families having one, two, three, four, and five children. To do this, we divide the number of families with a given number of children by the total number of families. This yields a simple distribution, with probabilities reflecting the likelihood of encountering families of various sizes in the sample. For practical applications and deeper understanding, graphical representations such as histograms or bar charts are often utilized to visualize the distribution.
Random Selection
Random selection refers to the process of choosing individuals or items in such a way that each individual or item has an equal chance of being selected. This concept is a cornerstone in probability theory and essential for conducting fair and unbiased statistical analyses.
In our exercise, random selection occurs twice, but in different contexts. Firstly, we randomly select a family out of all families, which exemplifies a random process where each family is equally likely to be chosen. Secondly, we select a child at random out of all children, which again, ensures that each child has the same opportunity to be picked. It is significant to note that although each family or child has an equal chance to be selected in their respective categories, the outcome probabilities are not the same due to the varied family sizes, highlighting the interesting interplay between random selection and conditional probability in shaping the final results.
In our exercise, random selection occurs twice, but in different contexts. Firstly, we randomly select a family out of all families, which exemplifies a random process where each family is equally likely to be chosen. Secondly, we select a child at random out of all children, which again, ensures that each child has the same opportunity to be picked. It is significant to note that although each family or child has an equal chance to be selected in their respective categories, the outcome probabilities are not the same due to the varied family sizes, highlighting the interesting interplay between random selection and conditional probability in shaping the final results.
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