Compound vs Conditional Probability
Clearing the confusion
Compound probability P(A∩B)
Probability of happening 2 events together (or in sequence).
Eg
- Probability of flipping a fair coin is ¹⁄₂ = 50%
- Probability of flipping 2 fair coins is ¹⁄₂ × ¹⁄₂ = ¹⁄₄ = 25% ( This can also be read as 50% of 50%)
P(A ∩ B) = P(A) ⋅ P(B)
For 2 coins, sample space would be 4 {HH,HT,TH,TT}, if the first coin is H then the remaining outcomes are 2 {HT,HH}. It means first event may impact the second event.
Eg
- Probability of picking 1 green ball from 10 different color balls is ¹⁄₁₀
- Probability of picking 2 green balls from 10 balls (2 green, 2 blue, 2 red, 4 yellow) ²⁄₁₀ × ¹⁄₉ (This will be more clear in permutation and combination)
In short. when the occurrence of 1st event impact the occurrence of 2nd event then they’re dependent events.
P(A ∩ B) = P(A) ⋅ P(B|A)
Here, P(B|A) is read as probability of B after A. This is the probability of happening B event when A event has already occurred. This is called conditional probability.
Compound probability vs Conditional probability
problem: A triangular region in a city is contaminated by a chemical industry. 2% of the children live in this triangle. 14% of these test positive for excessive toxic metals present in tissues. The rate of positive tests for children in the city not living in the triangle is only 1%.
Consider
- T denotes people living in triangular region, and
- P denotes people who were tested positive.
When it says that 14% of children in the triangle test positively, it means: if you draw a random child from the triangle, it will have a 14% chance of testing positive. This is P(P∣T)
The interpretation of P(P∩T) is the probability that a random person from the whole population is both in the triangle and tests positively.
Understanding with Venn diagram
P(A∩B) is the probability that both A and B have happened (without any additional information.)
P(A|B) is the probability that A has happened if we know B has happened.
Let’s understand it with an example. There is a class of 60 students. 33 like blue color, 23 like red, 20 students like both colors, and 4 students like orange color.
- What is the probability of selecting a student who like both colors.
No condition is given. So we’ll use universal sample space.
We’ll check the probability of selecting a student with particular choice from all students.
P(B ∩ R) = ²⁰⁄₆₀
2. What is the probability of selecting a student who like blue color 🄵🅁🄾🄼 the students who like red color
We’ll use sample space for given condition
We’ll check the probability of selecting a student with particular choice from particular set of students.
⇒ There are 23 students who like red color. Out of them 20 like both colors.
P(B | R) =²⁰⁄₂₃
With the Venn diagrams and above example we can say that in both situations, outcomes of desire event are not changing but the sample space is being reduced. Hence,
𝐏(𝐀∣𝐁) ≥𝐏(𝐀∩𝐁)
Examples
Let’s understand the difference with few more examples
Example 1
- Suppose you roll two dices, what is the probability of getting a six on the first and a four on the second?
- Suppose you roll two dices, what is the probability that the second die shows a four if the sum of the numbers on the two dice is ten?
In 1st case, there is no condition given to define the sample space. So we will take total outcomes possible from 2 dices i.e. 36. (Universal space)
P(A ∩ B) = 2/36
In 2nd case, there is a condition given for sample space i.e. the sample space of 2 numbers on dice which totals in 10. Total elements in sample space are only 3 {4+6,5+5,6+4}
P(A | B) = 1/3
Example 2
A person is crossing the street and we want to compute the probability when he gets hit by a passing car depending on the color of the traffic light.
Let H be whether the person gets hit or not, and C be the color of the traffic light.
- H={hit, not hit}
- C={red, yellow, green}.
The conditional probability that you get hit in this case is the probability P(H=hit|C=red), i.e. given that the light is red, how high is the chance that you will get hit by a car.
Well! there is a chance that a person can be hit even if the light is not red but here, we’re considering only the car accidents when light was red.
The compound (or joint) probability on the other hand, is P(H=hit, C=red), i.e. the probability that the light is red and you get hit by a car.
Suppose a person crossed the road 3 times without an accident. And 7 times he was hit by a car. In compound probability, we also want to know how many times the light was red when he was hit.
Now if we say, he crossed the road 10 times when the light was red and he was hit by the car 7 times. In such cases, condition for sample space is given.
Example 3
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:
Let’s find different probabilities;
- Find the probability that the student choose to fly as their superpower.
No condition for sample space is given. So we’ll take all students (100) to calculate the probability.
P(fly) = 38/100 = 0.38
2. Find the probability that the student was male.
Again there is no condition for sample space is given. So we’ll take all the students (100) to calculate the probability.
P(male) = 48/100 = 0.48
3. Find the probability that the student was male, 🄶🄸🅅🄴🄽 the student chose to fly as their superpower.
Now this is interesting, the sample space for this problem is the group of students who want to fly.
n(S) = 38
Out of 38 students, 26 are male. So
P(male ∣ fly) = 26/38 = 0.68
or using conditional probability formula
P(male ∩ fly) = Male students who chose to fly / Total students = 26/100
P(male ∣ fly) = P(male ∩ fly) / P(fly) = 26/38 = 0.68
4. Find the probability that the student chose to fly, given the student was male.
This is pretty much same as the last problem. Sample space for this problem would be n(S) = 48. Out of 48, there are 26 students who want to fly. So
P(male ∣ fly) = 26/48 =0.68
5. Let I represents the event where the student chose invisibility as their superpower, and F represents the event where the student was female.
Interpret the meaning of P(I ∣ F)≈0.62;
Select which option is correct;
a. About 62% of females chose invisibility as their superpower.
b. About 62% of people who chose invisibility as their superpower were female.
Explanation
- n(S) = All females
I ∣ Fcan be read as people out of all females who chose invisibility.
Together, we can read it as About 62% of females chose invisibility as their superpower. So statement (a) is correct.
Example 4
Shreya made the following table where she categorized countries by their region and by average business startup costs, as a percentage of the per capita gross national income (GNI) for a certain year.
Find the probability that the country is in the south Asia region, given that Shreya categorized the country’s business startup costs as high.
solution
This problem falls in conditional probability since a condition is given to select sample space.
- Countries where business startup costs is high n(S) = 87 (Sample space)
- Countries in South Asia region from above sample space i.e. the countries where business startup costs is high: 7
Probability of picking a country in South Asia region from countries where business startup costs is high = 7/87
If we use formula for conditional probability
We can first calculate P(A ∩ B) i.e. the probability of picking a country from all countries which is in South Asia region and where business startup costs is high.
There are 7 countries like this. Since there is no condition defined to select the sample space, we’ll take the universal space i.e. n(S) = 188.
P(A ∩ B) = 7/188
Now, we need to calculate the probability of a country where business startup costs is high. It is pretty simple
P(A) = 87/188
With the formula P(B|A) = 7/87
Disclaimer
All the examples I used to clear the confusion between compound and conditional probability are taken from different resources. Please check the reference section.
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