Define each of the following probability terms for the scenario of determining the probability that when a couple has 3 children, they will have exactly 2 boys. Assume that boys and girls are equally likely and that the gender of any child is not influenced by the gender of any other child.
a. Outcome
b. Set
c. Trial
d. Sample space
e. Event
Now you have done that, calculate the probability of the couple having 2 boys.
a. Outcome: In this scenario, an outcome refers to a specific result or combination of genders when the couple has 3 children. For example, one outcome could be having a boy, a boy, and a girl.
b. Set: A set refers to a collection of outcomes. In this case, a set could be all the possible outcomes when a couple has 3 children, including combinations such as girl, girl, girl or boy, boy, boy.
c. Trial: A trial is the process of performing a specific action or event that can have different outcomes. In this scenario, each time the couple has 3 children could be considered a trial.
d. Sample space: The sample space is the set of all possible outcomes for a specific event. In this case, the sample space would include all the possible combinations of genders the couple could have when they have 3 children.
e. Event: An event is a specific outcome or combination of outcomes that are of interest. In this scenario, the event of interest is having exactly 2 boys when the couple has 3 children.
To calculate the probability of the couple having exactly 2 boys, we need to determine the number of favorable outcomes (2 boys) and divide it by the total number of possible outcomes (sample space).
Number of favorable outcomes = 3 (boy, boy, girl or girl, boy, boy or boy, girl, boy)
Sample space = 2^3 = 8 (as there are 2 possibilities for each child - boy or girl)
Therefore, the probability of the couple having exactly 2 boys is 3/8, or 37.5%.
a. Outcome: In probability, an outcome refers to a possible result of an experiment or event. In this scenario, the possible outcomes are the different combinations of genders the couple's three children can have.
b. Set: A set is a collection of distinct elements. In probability, a set is often used to represent a specific condition or criteria. In this scenario, a set could be defined as the set of outcomes where the couple has exactly 2 boys.
c. Trial: A trial is a single performance of an experiment or event. In this scenario, a trial would be considered as one instance of a couple having 3 children.
d. Sample space: The sample space refers to the set of all possible outcomes of an experiment or event. In this scenario, the sample space would be the set of all possible combinations of genders the couple's three children can have, considering that boys and girls are equally likely. The sample space would be {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG}, where the letters B and G represent boys and girls respectively.
e. Event: In probability, an event is a specific outcome or a set of outcomes of an experiment or event. It represents a particular occurrence or condition. In this scenario, the event would be defined as the event of the couple having exactly 2 boys, i.e., BGG, GBG, or GGB.
To calculate the probability of the couple having exactly 2 boys, we need to determine the ratio of the number of outcomes in the event to the total number of outcomes in the sample space.
The event consists of 3 outcomes (BGG, GBG, GGB), while the sample space consists of 8 outcomes.
Therefore, the probability of the couple having 2 boys is 3/8, or 0.375, which can also be expressed as 37.5%.
a. Outcome: The outcome refers to a specific result or possibility that can occur in an experiment or scenario. In this case, each child can be either a boy or a girl, so there are two possible outcomes for each child.
b. Set: A set is a collection of distinct outcomes. In this scenario, one possible set could be the collection of outcomes where the couple has exactly two boys and one girl.
c. Trial: A trial refers to a single occurrence or attempt in an experiment or scenario. In this case, a trial would be the act of having one child.
d. Sample space: The sample space is the set of all possible outcomes in an experiment or scenario. In this case, since each child can be either a boy or girl, the sample space would consist of all the possible combinations of boys and girls that can be born in a family with three children.
The sample space for the scenario of determining the probability of having exactly 2 boys when a couple has 3 children can be represented by the following outcomes:
BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG
e. Event: An event refers to a specific collection of outcomes from the sample space. In this case, the event would be the collection of outcomes where the couple has exactly 2 boys.
To calculate the probability of the couple having 2 boys, we need to determine the number of outcomes in the event of interest (having exactly 2 boys) and divide it by the total number of outcomes in the sample space.
In this case, there are 3 possible outcomes in the event of having exactly 2 boys: BBG, BGB, and GBB. The sample space consists of 8 possible outcomes.
So, the probability of the couple having exactly 2 boys is 3/8 or 0.375, which can also be expressed as 37.5%.