Suppose a random family of four is selected (two parents and two children). Suppose further that you are told one of the two children is a girl. Given that this family has at least one daughter, what is the conditional probability that it also has a son? (HINT: write out the sample space and then use your knowledge of "conditional" probabilities)
To solve this problem, we can use the concept of conditional probability.
Let's first write out the sample space for the possible genders of the two children in the family:
Sample Space: {BB, BG, GB, GG}
- BB: Both children are boys
- BG: One child is a boy, and the other is a girl
- GB: One child is a girl, and the other is a boy
- GG: Both children are girls
Based on the information given that at least one of the children is a girl, we can eliminate the BB outcome from the sample space, as it is not possible. This leaves us with the reduced sample space:
Reduced Sample Space: {BG, GB, GG}
Now, we need to determine the probability of the family having a son given that they have at least one daughter. This can be calculated using the conditional probability formula as follows:
Conditional Probability = (Desired Outcome) / (Total Reduced Sample Space)
The desired outcome is the number of outcomes where the family has a son, which is GB. The total reduced sample space is the number of outcomes in the reduced sample space, which is 3 (BG, GB, GG).
Hence, the conditional probability is:
Conditional Probability = 1 (GB) / 3 (BG, GB, GG)
= 1 / 3
Therefore, the conditional probability that the family has a son given that they have at least one daughter is 1/3.
To solve this problem, we need to use conditional probability. Conditional probability is the probability of an event happening given that another event has already occurred. In this case, we need to find the probability of the family having a son given that they have at least one daughter.
Let's write out the sample space to understand the different possibilities:
- Case 1: Older child is a girl, younger child is a girl (GG)
- Case 2: Older child is a girl, younger child is a boy (GB)
- Case 3: Older child is a boy, younger child is a girl (BG)
- Case 4: Older child is a boy, younger child is a boy (BB)
Since we are told that at least one child is a girl, we can eliminate case 4 (BB) from our sample space. Now we are left with three possibilities: GG, GB, and BG.
Given that one child is a girl, we have two remaining possibilities where the family has at least one daughter: GG and GB. Out of these two possibilities, only GB has a son.
Therefore, the favorable outcome (family having a son) is GB, and the total possible outcomes (family having at least one daughter) include GG and GB.
So, the conditional probability of the family having a son given that they have at least one daughter is 1 out of 2, or 1/2.
To summarize: The conditional probability that the family has a son, given that they have at least one daughter, is 1/2.