We are given the following statistics about the number of children in the families of a small village.
There are 100 families: 10 families have no children, 40 families have 1 child each, 30 families have 2 children each, 10 families have 3 each, and 10 families have 4 each.
1. If you pick a family at random (each family in the village being equally likely to be picked), what is the expected number of children in that family?
- unanswered
2. If you pick a child at random (each child in the village being equally likely to be picked), what is the expected number of children in that child's family (including the picked child)?
- unanswered
3. Generalize your approach from part 2: Suppose that a fraction of the families have children each. Let be the number of children in a randomly selected family, and let and . Let be the number of children in a randomly chosen child's family. Express in terms of and using standard notation.
- Let x be the fraction of families with n children each, and let y be the number of children in a randomly selected family, and let a and b be the number of children in a randomly chosen child's family. Then b = y + 1.
1. To find the expected number of children in a randomly chosen family, we can calculate the weighted sum of the number of children in each family.
There are 10 families with no children, 40 families with 1 child, 30 families with 2 children, 10 families with 3 children, and 10 families with 4 children.
Expected number of children in a family = (0 * 10 + 1 * 40 + 2 * 30 + 3 * 10 + 4 * 10) / 100
= (0 + 40 + 60 + 30 + 40) / 100
= 170 / 100
= 1.7
Therefore, the expected number of children in a randomly chosen family is 1.7.
2. To find the expected number of children in a randomly chosen child's family (including the picked child), we need to take into account the distribution of children in the families.
Out of the 100 families in the village, there is a total of 1 * 40 + 2 * 30 + 3 * 10 + 4 * 10 = 40 + 60 + 30 + 40 = 170 children.
Expected number of children in a child's family = (1 * 40 + 2 * 30 + 3 * 10 + 4 * 10) / (1 * 40 + 2 * 30 + 3 * 10 + 4 * 10 + 170)
= (40 + 60 + 30 + 40) / (40 + 60 + 30 + 40 + 170)
= 170 / 340
= 0.5
Therefore, the expected number of children in a randomly chosen child's family (including the picked child) is 0.5.
3. Let n be the number of children in a randomly chosen family, m be the number of children in a randomly chosen child's family, p be the fraction of families with m children each, and q be the expected number of children in a family.
The fraction of families with m children each is p.
Therefore, the fraction of families with n children each is also p, because both n and m are chosen randomly from the same distribution.
The expected number of children in a family with n children is q.
Therefore, we can express p in terms of q and m as p = q * pm(m), where pm(m) is the probability of a family having m children.
Similarly, we can express q in terms of p and n as q = (1 / pn(n)), where pn(n) is the probability of a family having n children.
Hence, the expression for m in terms of p and q is m = (1 / pm(m)) * (1 / pn(n)).
(Note: The specific form of the probability distributions pm(m) and pn(n) would need to be given to obtain a more precise expression for m).
To find the expected number of children in a family, we will use the concept of probability.
1. To find the expected number of children in a randomly chosen family:
- We have 100 families in total.
- 10 families have no children, so the probability of selecting a family with no children is 10/100 = 0.1.
- 40 families have 1 child each, so the probability of selecting a family with 1 child is 40/100 = 0.4.
- 30 families have 2 children each, so the probability of selecting a family with 2 children is 30/100 = 0.3.
- 10 families have 3 children each, so the probability of selecting a family with 3 children is 10/100 = 0.1.
- 10 families have 4 children each, so the probability of selecting a family with 4 children is 10/100 = 0.1.
To calculate the expected number of children, we multiply the number of children in each category by their respective probabilities and sum them up:
Expected number of children = (0 * 0.1) + (1 * 0.4) + (2 * 0.3) + (3 * 0.1) + (4 * 0.1)
Expected number of children = 0 + 0.4 + 0.6 + 0.3 + 0.4
Expected number of children = 1.7
Therefore, the expected number of children in a randomly chosen family is 1.7.
2. To find the expected number of children in a child's family:
- We can start by calculating the probability of selecting a child from each category of families.
- The probability of picking a child from a family with no children is 0, as there are no children in those families.
- The probability of picking a child from a family with 1 child is (1/100) * (40/100) = 0.004.
- The probability of picking a child from a family with 2 children is (2/100) * (30/100) = 0.006.
- The probability of picking a child from a family with 3 children is (3/100) * (10/100) = 0.003.
- The probability of picking a child from a family with 4 children is (4/100) * (10/100) = 0.004.
To calculate the expected number of children in a child's family, we multiply the number of children in each category by their respective probabilities and sum them up:
Expected number of children in a child's family = (1 * 0.004) + (2 * 0.006) + (3 * 0.003) + (4 * 0.004)
Expected number of children in a child's family = 0.004 + 0.012 + 0.009 + 0.004
Expected number of children in a child's family = 0.029
Therefore, the expected number of children in a child's family (including the picked child) is 0.029.
3. To generalize the approach from part 2:
Let p_i be the probability of selecting a family with i children, where i ranges from 0 to n.
To find the expected number of children in a child's family (including the picked child), we need to calculate the weighted average.
Expected number of children in a child's family = (1 * p_1) + (2 * p_2) + ... + (n * p_n)
We can express this using sigma notation:
Expected number of children in a child's family = Σ(i * p_i), where i ranges from 1 to n.
Here, p_i is the probability of selecting a family with i children, and i is the number of children in the family.
So, in general:
The expected number of children in a child's family is given by: Σ(i * p_i), where i ranges from 1 to n.
Using this notation, we can express the expected number of children in a child's family from part 2 as: Σ(i * p_i), where i ranges from 1 to 4.