Wize University Statistics Textbook > Probability

Probability of Dependent Events (Conditional Probability)

Probability of Dependent Events (Conditional Probability)

Events A and B are dependent if the occurrence of one has an affect on the probability of the occurrence of the other.

Conditional Probability

The probability that event A will occur given that event B has already occured is called the "conditional probability of event A given event B", and is denoted by $P(A|B)$ .

Note: $P(B|A)$ is the probability that event B will occur given that event A already occured.

Multiplication Rule

For any two events A and B, we have the following multiplication rules:

Image illustrating Probability of Dependent Events (Conditional Probability)

We can rewrite these rules by isolating for the conditional probability, giving us the following formulas for conditional probability:

Example

There are 6 possible outcomes of a fair die. Sample space $S=\left\{1,2,3,4,5,6\right\}$

$A=\ \left\{4,5,6\ \right\}$

$B=\ \left\{1,3,5\right\}$

Find $P\left(A\mid B\right)$ .

There is only 1 outcome {5} that are in both A and B out of the sample space of 6 possible outcomes {1,2,3,4,5,6}

So, $P\left(A\cap B\right)=\frac{1}{6}$

There are 3 outcomes {1,3,5} that are in the sample space of 6 possible outcomes {1,2,3,4,5,6}

So, $P\left(B\right)=\frac{3}{6}$

Therefore,

$\displaystyle P\left(A\mid B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{\frac{1}{6}}{\frac{3}{6}}=\frac{1}{3}$

Conditional Probability: Tree Diagram

$P\left(A\mid B\right)\ =$ the conditional probability that event A will occur given that event B already occurred.

$P\left(A\mid B^c\right)\ =$ the conditional probability that event A will occur given that event B did not occur.

As you can see, event A could occur in more than one way, depending on whether event B occurs or not. The probability that event A will occur is therefore the sum of all the outcomes where Event A occurs:

Image illustrating Conditional Probability: Tree Diagram

Similarly, event B could occur more than one way, depending on whether event A occurs or not. The probability that event B will occur is the sum of all the outcomes where Event B occurs:

Tree Diagram

A tree diagram shows all the possible outcomes and can help you visualize the problem. It consists of branches that are labeled with probabilities.

Suppose A and B are two dependent events, the following tree diagram shows all the possible outcomes:

Notice that Event B occurs twice:

given that Event A occurs; and
given that Event Ac occurs.

Therefore, the probability of Event B occurring is the sum of the two probabilities that Event B occurs:

\boxed{P(B)=P(B|A)\cdot P(A)+P(B|A^c)\cdot P(A^c)}

Bayes’ Theorem

Bayes’ Theorem describes the probability of an event based on conditions that might be related to the event. In other words, suppose we already know the outcome is true and wish to infer the prior probability of the event.

Examples

Mr. Smith is diagnosed with appendicitis. What’s the probability that he is older than 30 years old?
Julia agreed to a second date with Richard. What’s the probability that he paid for dinner?
Paul adopted a cat. What’s the probability that the cat is black?

Bayes' Theorem formula

Example

Find P(A|B):

$\displaystyle P(A|B)=\frac{P(B|A)P(A)}{P\left(B\right)}=\frac{P(B|A)P(A)}{P(B\ |A)P(A)+P(B\ |A^c)+P(B\ |A^c)P(A^c)}=\frac{0.05}{0.05+0.15}=0.25$

Example: Bayes' Theorem

$4\%$ of dogs have an infection. Of those with an infection, $10\%$ are German Pinschers (GP). $1\%$ of those that aren't infected are German Pinschers.

Tree diagram:

(a) If we randomly select a dog, what is the probability that it's a German Pinscher?

$P\left(GP\right)=\left(0.04\right)\left(0.10\right)+\left(0.96\right)\left(0.01\right)=0.0136$

(b) Phoebe owns a German Pinscher. What is the probability that her dog is infected?

Bayes' Theorem:

$\displaystyle P\left(I\ |\ GP\right)$

$\displaystyle =\frac{P\left(I\ \cap GP\right)}{P\left(GP\right)}$

$\displaystyle =\frac{P\left(I\right)P\left(GP\mid I\right)}{P\left(I\right)P\left(GP\mid I\right)+P\left(I^c\right)P\left(GP\mid I^c\right)}$

$\displaystyle =\frac{\left(0.04\right)\left(0.10\right)}{\left(0.04\right)\left(0.10\right)+\left(0.96\right)\left(0.01\right)}$

$\displaystyle =\frac{0.004}{0.0136}$

$\approx0.2941$

Alternatively, you can use the tree diagram instead of the Bayes' Theorem formula:

$\displaystyle P(I|GP)=\frac{P(I\cap GP)}{P(GP)}$

We can find the numerator by following the I and GP branches: $P(I\cap GP)=0.04\times0.10=0.004$
We already found the denominator in part a): $P(GP)=0.0136$

Therefore, the probability is $\displaystyle P(I|GP)=\frac{0.004}{0.0136}\approx0.2941$

Katie could put a ball in either Slot A or Slot B and hopes to win a prize. The ball will bounce around and will either land in a winning box or the losing box. Suppose she puts the ball in Slot A 40% of the time; Slot B 60% of the time. If she puts the ball in Slot A, the ball lands in the winning box 15% of the time. If she puts the ball in Slot B, the ball lands in the losing box through the first channel 60% of the time and into the same losing box through the second channel 20% of the time. See illustration below:

Illustration for practice question: Bayes' Theorem

Katie won. What is the probability that she put the ball in Slot A?

0.06

0.12

0.18

0.33

I don't know

Extra Practice

Probability of Dependent Events: Conditional Probability

Let A and B be two events where P (A) = 0.7 and P (B) = 0.5. Which of the following is false?

Probability of dependent events: Conditional probability

Mizter Chi raps about statistics: 30% about descriptive statistics and 70% about inferential statistics. Based on his songs, 62% of them contains profanity, but 80% of his songs about inferential statistics contains profanity. What percentage of his songs about descriptive statistics does not contain profanity?

Probability of Dependent Events

Half of students own an Apple product. One-quarter of all students wear glasses. 35% of all students do not wear glasses and nor own Apple products.

Use the contingency table from the previous question to answer the following questions below. [Click on Hint to see the full contingency table.]

Bayes' Theorem: Conditional Probability

Katie won. What is the probability that she put the ball in Slot A?

Probability for Dependent Events

Given that P(A) = 0.6, P(B) = 0.5, and P(Bc|Ac)= 0.8, create a contingency table to answer the following questions.

Probability for Dependent Events 2

Gloria is an opera singer. The probability that she will get a standing ovation is 60%. The probability that she will have a wardrobe malfunction is 20%. The probability that she will only get a standing ovation is 55%.

Probability of Dependent Events (Conditional Probability)

The probability that a first year student is enrolled in BMOS is 0.40. 30% of all BMOS students take calculus and 20% of students not in BMOS take calculus.

Two-Way Table Practice Question

40% of children eat strawberries. Of those that eat strawberries, 80% are vegetarian. Half of those that don’t eat strawberries are vegetarian.

Conditional Probability

A blue and red die are tossed at the same time. If it’s known that the sum of the two dice is $9$ , what is the probability that the blue die shows a $4$ ?

Conditional Probability

A certain pregnancy test is 98% effective in detecting an actual pregnancy. However, the test gives a "false positive" result for 2% of the non-pregnant women tested (i.e. if a woman is not pregenent and tries the test, the probability that the test will indicate pregnancy is 0.02). If 20% of women in a particular group are in fact pregnant, what is the probability that a women selected at random from this group is pregnant if the result is positive?

Probability of Dependent Events (Conditional Probability)

An on-campus survey asked 200 students where their home town is, and here is the frequency distribution of their answers.

$\begin{array}{|c|c|c|c|c|} \hline \text{Home Town}&\text{Calgary}&\text{Toronto}&\text{Montreal}&\text{Vancouver}\\ \hline \text{Frequency}&80&75&15&30\\ \hline \end{array}$

If a randomly selected student told us that he/she is not from Toronto, what is the probability that he/she is from Calgary?

Probability of Dependent Events (Conditional probability)

This table shows whether passengers on a flight to New York like to fly and/or like to drive.

Let F be the event that a randomly selected passenger likes to fly.
Let D be the event that a randomly selected passenger likes to drive.

Bernoulli Trials & Conditional Probability

Four teams are competing in a hot dog eating contest where they must finish eating 20 hot dogs in the given amount of time. For each team, the probability of them finishing the contest is 0.6 If at least one team finished the contest, determine the probability that all Four of them did.

Conditional Probability

Probability of Dependent Events (Conditional Probability)

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