Examples of conditional probability in the following topics:
-
- We call this a conditional probability because we computed the probability under a condition: parents = used.
- There are two parts to a conditional probability, the outcome of interest and the condition.
- Thus, the conditional probability could be computed:
- The fraction of these probabilities represents our general formula for conditional probability.
- The conditional probability of the outcome of interest A given condition B is computed as the following:
-
- The conditional probability of an event is the probability that an event will occur given that another event has occurred.
- The notation $P(B|A)$ indicates a conditional probability, meaning it indicates the probability of one event under the condition that we know another event has happened.
- The conditional probability $\displaystyle P(B|A)$ of an event $B$, given an event $A$, is defined by:
- The conditional probability $P(B|A)$ is not always equal to the unconditional probability $P(B)$.
- Mathematically, Bayes' theorem gives the relationship between the probabilities of $A$ and $B$, $P(A)$ and $P(B)$, and the conditional probabilities of $A$ given $B$ and $B$ given $A$.
-
- We can show this is mathematically true using conditional probabilities.
- (a) What is the probability that the first die, X, is 1?
- (b) What is the probability that both X and Y are 1?
- (c) Use the formula for conditional probability to compute P(Y = 1 |X = 1).
- (d) The probability is the same as in part (c): P(Y = 1) = 1/6.
-
- The multiplication rule states that the probability that $A$ and $B$ both occur is equal to the probability that $B$ occurs times the conditional probability that $A$ occurs given that $B$ occurs.
- In probability theory, the Multiplication Rule states that the probability that $A$ and $B$ occur is equal to the probability that $A$ occurs times the conditional probability that $B$ occurs, given that we know $A$ has already occurred.
- We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator.
- The probability that we get a $2$ on the die and a tails on the coin is $\frac{1}{6}\cdot \frac{1}{2} = \frac{1}{12}$, since the two events are independent.
- Apply the multiplication rule to calculate the probability of both $A$ and $B$ occurring
-
- In many instances, we are given a conditional probability of the form:
- But we would really like to know the inverted conditional probability:
- Tree diagrams can be used to find the second conditional probability when given the first.
- In the example, each of the probabilities on the right side was broken down into a product of a conditional probability and marginal probability using the tree diagram.
- Consider the following conditional probability for variable 1 and variable 2:
-
- Tree diagrams are annotated with marginal and conditional probabilities, as shown in Figure 2.17.
- The secondary branches are conditioned on the first, so we assign conditional probabilities to these branches.
- For example, the top branch in Figure 2.17 is the probability that result = lived conditioned on the information that inoculated = yes.
- To calculate this conditional probability, we need the following probabilities:
- The final grades, which correspond to the conditional probabilities provided, will be shown on the secondary branches.
-
- The probability distribution of a discrete random variable $x$ lists the values and their probabilities, where value $x_1$ has probability $p_1$, value $x_2$ has probability $x_2$, and so on.
- Every probability $p_i$ is a number between 0 and 1, and the sum of all the probabilities is equal to 1.
- Of the conditional probabilities of the event $B$ given that $A_1$ is the case or that $A_2$ is the case, respectively.
- The formula, table, and probability histogram satisfy the following necessary conditions of discrete probability distributions:
- The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable.
-
- See the discussion on conditional probabilities on this page to see how to compute P(A and B) when A and B are not independent.
- Once the first card chosen is an ace, the probability that the second card chosen is also an ace is called the conditional probability of drawing an ace.
- In this case, the "condition" is that the first card is an ace.
- If your first thought is that it is 25/365 = 0.068, you will be surprised to learn it is much higher than that.This problem requires the application of the sections on P(A and B) and conditional probability.
- Now define P3 as the probability that the third person drawn does not share a birthday with anyone drawn previously given that there are no previous birthday matches.P3 is therefore a conditional probability.
-
- These totals represent marginal probabilities for the sample, which are the probabilities based on a single variable without conditioning on any other variables.
- For instance, a probability based solely on the student variable is a marginal probability:
- If a probability is based on a single variable, it is a marginal probability.
- Verify Table 2.14 represents a probability distribution: events are disjoint, all probabilities are non-negative, and the probabilities sum to 1.24.
- We can compute marginal probabilities using joint probabilities in simple cases.
-
- The hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws without replacement from a finite population of size $N$ containing a maximum of $K$ successes.
- The following conditions characterize the hypergeometric distribution:
- In the softball example, the probability of picking a women first is $\frac{13}{24}$.
- The probability of picking a man second is $\frac{11}{23}$, if a woman was picked first.
- The probability of the second pick depends on what happened in the first pick.