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.
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
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
Generalize your approach from part 2: Suppose that a fraction pk of the families have k children each. Let K be the number of children in a randomly selected family, and let a=E[K] and b=E[K2]. Let W be the number of children in a randomly chosen child's family. Express E[W] in terms of a and b using standard notation.
E[W]=
- unanswered
b) 41/17
1) 17/10
2)41/10
3) b/a
2) the correct answer is
41/17
E[W]= Do you know what E stands for?
To find the expected number of children in a family, we can use the concept of expected value. The expected value is calculated by multiplying each possible outcome by its probability of occurring and summing them up.
For the given statistics, we have:
- 10 families with no children (0 children each)
- 40 families with 1 child each
- 30 families with 2 children each
- 10 families with 3 children each
- 10 families with 4 children each
To calculate the expected number of children in a family, we multiply the number of children in each category by the probability of selecting a family from that category and sum them up.
Expected number of children = (0 * 10/100) + (1 * 40/100) + (2 * 30/100) + (3 * 10/100) + (4 * 10/100)
= (0 + 40/100 + 60/100 + 30/100 + 40/100)
= (0 + 0.4 + 0.6 + 0.3 + 0.4)
= 2.1
Therefore, the expected number of children in a randomly selected family is 2.1.
Now, let's calculate the expected number of children in a child's family. Since each child is equally likely to be picked, we need to consider the probability of a child belonging to a family with a certain number of children.
To find this, we divide the number of families with a certain number of children by the total number of families. Let's denote the number of children in a randomly selected child's family as W.
For W = 0:
Probability(W = 0) = Number of families with 0 children / Total number of families = 10/100 = 0.1
For W = 1:
Probability(W = 1) = Number of families with 1 child / Total number of families = 40/100 = 0.4
Similarly, we can calculate the probabilities for W = 2, 3, and 4.
Expected number of children in a child's family (E[W]) = (0 * 0.1) + (1 * 0.4) + (2 * 0.3) + (3 * 0.1) + (4 * 0.1)
= (0 + 0.4 + 0.6 + 0.3 + 0.4)
= 2.1
So, the expected number of children in a randomly selected child's family is also 2.1.
Now, let's generalize this approach for any given fraction pk of families having k children each and calculate the expected number of children in a randomly chosen child's family using the variables a and b.
E[W] = (0 * p0) + (1 * p1) + (2 * p2) + ... + (k * pk)
Note: Here, p0 + p1 + p2 + ... + pk = 1 (because the sum of probabilities should be equal to 1).
Therefore, E[W] = p1 + 2p2 + 3p3 + ... + kpk
This notation is commonly used to express the expected value of a random variable.