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### Course: High school statistics>Unit 6

Lesson 3: Conditional probability

# Conditional probability using two-way tables

Researchers surveyed $100$ students on which superpower they would most like to have. This two-way table displays data for the sample of students who responded to the survey:
SuperpowerMaleFemaleTOTAL
Fly$26$$12$$38$
Invisibility$12$$32$$44$
Other$10$$8$$18$
TOTAL$48$$52$$100$
A student will be chosen at random.
problem 1
Find the probability that the student chose to fly as their superpower.
$P\left(\text{fly}\right)=$

problem 2
Find the probability that the student was male.
$P\left(\text{male}\right)=$

problem 3
Find the probability that the student was male, given the student chose to fly as their superpower.

problem 4
Find the probability that the student chose to fly, given the student was male.

problem 5
Is this statement about conditional probability true or false?
"In general, . You can reverse the order and the probability is the same either way."

problem 6
Let $\text{I}$ represent the event where the student chose invisibility as their superpower, and $\text{F}$ represent the event where the student was female.
Interpret the meaning of .

## Want to join the conversation?

• How did you get your answer that 62% of females chose invisibility as their superpower. I saw up above, it was 44.
• 44 is the TOTAL number of people who chose invisibility. Out of those, 32 are female, therefore 32 is the condition that satisfies our probability question (the numerator in the probability formula).

52 is the total number of people who are female in this experiment.

32/52 is about 0.62 or 62%
• I might need to practice this more at home and to read my notes more carefully. It's not easy, but I'll take it as a challenge!
• I think It is easy to you now, after 5 years.
• What is the quickest way to calculate probability?
• in a bag with six things in which two things are pens, what is the probability of you hitting a pen by putting your hand on the bag without looking at it's inside?

you just need to divide the number of pens from the number of things, that is gonna be 2/6 # two pens divided by six things, or you can simplify and you get 1/3, so you have 1/3 probability of hitting a pen.
(1 vote)
• Is there any formula for conditional probability, or is it simply common sense? (in general questions)
• In general, if A and B are events such that P(B) is nonzero, then
P(A given B) = P(A and B) / P(B).

Have a blessed, wonderful day!
• I honestly don't understand this, Thank you for making all this content available.
• Are there harder ways to do this type of question?
• I think Bayes' Theorem questions can get a lot harder than this. We're being given all the information here, so it's easy to calculate any kind of probability, but it gets harder when you don't have all the information and you have to extrapolate.
• Why shouldn't we apply Bayes theorem in question 3 (P(male ∣ fly))?
• Bayes' Theorem says
P(male | fly) = P(fly | male) ∙ P(male)∕P(fly)

But we don't know what P(fly | male) is, so we can't use this formula.
• the last one means the same thing to me idk maybe im illiterate
(1 vote)
• Well, A is stating that 62% of women chose invisibility while B says that 62% of both men and women who CHOSE Invisibility are female which would be: P(F given I) not P(I given F)
• When will I have to find the overlap between events and when would I have to find the total
• Finding the overlap between events: You do this when you're interested in the probability of two conditions both being true at the same time. For example, finding the probability that a randomly chosen student is both male and chose to fly as their superpower. This is often represented as P(A and B) and involves looking at the specific intersection in a two-way table where those conditions meet.

Finding the total: This is necessary when you're calculating the probability of a single condition without concern for a second condition, or when you're calculating probabilities that involve the total number of outcomes. For example, finding the probability of a student choosing to fly or the probability of a student being male involves summing over the relevant categories to get the total count for that condition, then dividing by the grand total of all observations.
(1 vote)
• 1-what is the difference between P(I ∣ F) and P(F ∣ I) when I Interpret the meaning?
2-what the difference in meaning between P(I ∣ F) and
P(I and F) ?
can anyone help me to understand these questions?
(1 vote)
• Difference is similar to this:
"If I will get home I will sleep" or "If I will sleep I will get home"

Writing P(A | B) is "If B happens, how likely is A to happen".
Writing P(B | A) is "If A happens, how likely is B to happen".