1. Probabilities with Formulas On the last writing homework, we used diagrams to compute several probabilities about the US Congress. On this writing homework, we will use formulas to compute more probabilities. Useful Information As before, our probability experiment is to choose one congress member at random. Let A be the event that they are Republican. Let B be the event that they are Male. Let C be the event that they are Caucasian. Here are some probabilities we know from last time that you can use in your computations below: P(A’) = 0.54 P(B’) = 0.24 P(C) = 0.72 P( B u C ) = 0.89 P( A n B ) = 0.42 P( A|B ) = 0.56 P( B|A ) = 0.93 P( C|A ) = 0.92 Here are the formulas we know from class: Addition Rule: P( M u N ) = P(M) + P(N) - P( M n N ) Multiplication Rule: P( M n N ) = P(M | N) * P(N) = P(N | M) * P(M) Complement Rule: P(M) + P( M’ ) = 1 Your task For each of the following probabilities: Translate either from symbols to words or words to symbols. Clearly identify the formula you will use to compute the probability. Plug in the values into the formula. The values can come from above or can come from what you have computed already on this problem. For example: “Compute the probability that a randomly chosen member of congress is both Male and Caucasian.” In symbols, this means P( B n C ). I will use the Addition formula, P( B u C ) = P(B) + P(C) - P(B n C). From info given above, I know that P( B u C ) = 0.89 and P(C) = 0.72. Below you will compute P(B). Plug these numbers in, then solve for P(B n C). Typos fixed Compute P(B) (Hint: use the complement rule.) Compute P( B n C ) (Finish the computation from above.) Compute P( B | C ) (Hint: use P(C), P( B n C ) and the multiplication rule.) Compute P( C | B ) (Hint: use P(B), P( B n C ) and the multiplication rule.) Compute the probability that a randomly chosen member of congress is Republican. Compute the probability that a randomly chosen member of congress is both Republican and Caucasian. Compute the probability that a randomly chosen member of congress is Republican, given that they are Caucasian.
Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
Tree diagram
Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
Writing Prompts
1. Probabilities with Formulas
On the last writing homework, we used diagrams to compute several probabilities about the US Congress. On this writing homework, we will use formulas to compute more probabilities.
Useful Information
As before, our probability experiment is to choose one congress member at random.
- Let A be the
event that they are Republican. - Let B be the event that they are Male.
- Let C be the event that they are Caucasian.
Here are some probabilities we know from last time that you can use in your computations below:
|
P(A’) = 0.54 |
P(B’) = 0.24 |
P(C) = 0.72 |
P( B u C ) = 0.89 |
|
P( A n B ) = 0.42 |
P( A|B ) = 0.56 |
P( B|A ) = 0.93 |
P( C|A ) = 0.92 |
Here are the formulas we know from class:
Addition Rule : P( M u N ) = P(M) + P(N) - P( M n N )- Multiplication Rule: P( M n N ) = P(M | N) * P(N) = P(N | M) * P(M)
- Complement Rule: P(M) + P( M’ ) = 1
Your task
For each of the following probabilities:
- Translate either from symbols to words or words to symbols.
- Clearly identify the formula you will use to compute the probability.
- Plug in the values into the formula. The values can come from above or can come from what you have computed already on this problem.
For example:
“Compute the probability that a randomly chosen member of congress is both Male and Caucasian.”
In symbols, this means P( B n C ).
I will use the Addition formula, P( B u C ) = P(B) + P(C) - P(B n C).
From info given above, I know that P( B u C ) = 0.89 and P(C) = 0.72. Below you will compute P(B). Plug these numbers in, then solve for P(B n C).
Typos fixed
- Compute P(B) (Hint: use the complement rule.)
- Compute P( B n C ) (Finish the computation from above.)
- Compute P( B | C ) (Hint: use P(C), P( B n C ) and the multiplication rule.)
- Compute P( C | B ) (Hint: use P(B), P( B n C ) and the multiplication rule.)
- Compute the probability that a randomly chosen member of congress is Republican.
- Compute the probability that a randomly chosen member of congress is both Republican and Caucasian.
- Compute the probability that a randomly chosen member of congress is Republican, given that they are Caucasian.
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