Mathematics 03: Conditional Probability

Conditional probability means:

What is the chance of A when we already know B happened?

The words given that are the most important clue.

The class question

You receive a tuple in this order:

(total students, passed Math, passed both Math and Science)

Write a Python program that finds:

What is the probability that a student passed Science given that the student passed Math?

The output must be rounded to two decimal places. Invalid inputs must print Invalid input.

Example input

(20, 5, 5)

Example output

1

1. Name the values first

Let:

The input tuple is (T, M, B).

For (20, 5, 5):

T = 20
M = 5
B = 5

2. Basic probability formula

Question — What is the chance that a randomly chosen student passed Math?

Formula

P(A) = number of favourable outcomes / total number of outcomes

For Math:

P(M) = M / T

For (20, 5, 5):

P(M) = 5 / 20 = 0.25

So 25% of all students passed Math.

3. Intersection probability

The symbol ∩ means and or both.

M ∩ S = passed Math and Science

Formula

P(M ∩ S) = B / T

For (20, 5, 5):

P(M ∩ S) = 5 / 20 = 0.25

So 25% of all students passed both subjects.

4. Conditional probability

We do not ask about every student in the class anymore. We narrow the group to only students who passed Math.

The question is:

P(S | M)

Read this as:

Probability of passing Science given that the student passed Math.

Conditional probability formula

P(A | B) = P(A ∩ B) / P(B), where P(B) > 0

For this question:

P(S | M) = P(S ∩ M) / P(M)

Substitute the formulas from the previous sections:

P(S | M) = (B / T) / (M / T)

Dividing by a fraction means multiplying by its reciprocal:

P(S | M) = (B / T) × (T / M)

T cancels:

P(S | M) = B / M

Final program formula

P(Science | Math) = passed both Math and Science / passed Math
                   = B / M

In plain English:

From the students who passed Math, find the fraction who also passed Science.

The total T is not needed for the final division. It is still important for validating that the input describes a possible class.

Worked example 1 — (20, 5, 5)

M = 5
B = 5

P(S | M) = B / M
         = 5 / 5
         = 1

Output:

1

Every student who passed Math also passed Science. 1 means 100%.

Worked example 2 — (20, 5, 2)

M = 5
B = 2

P(S | M) = 2 / 5
         = 0.4

Output:

0.4

Among Math-passing students, 40% also passed Science.

Worked example 3 — (20, 8, 3)

P(S | M) = 3 / 8
         = 0.375

Rounded to two decimal places:

0.38

5. Why (20, 5, 10) is invalid

This input says:

5 students passed Math
10 students passed both Math and Science

This is impossible. Everyone who passed both subjects must be included in the students who passed Math.

B ≤ M

But here:

10 > 5

So the correct output is:

Invalid input

6. Input validation equations

A valid input must satisfy all of these rules:

T > 0
0 < M ≤ T
0 ≤ B ≤ M

7. Python solution

import ast

try:
    data = ast.literal_eval(input())

    if not isinstance(data, tuple) or len(data) != 3:
        print("Invalid input")
    else:
        total, math_passed, both_passed = data

        valid_counts = all(isinstance(value, int) and not isinstance(value, bool) for value in data)
        valid_relationships = (
            total > 0
            and math_passed > 0
            and math_passed <= total
            and both_passed >= 0
            and both_passed <= math_passed
        )

        if not valid_counts or not valid_relationships:
            print("Invalid input")
        else:
            probability = both_passed / math_passed
            result = f"{probability:.2f}".rstrip("0").rstrip(".")
            print(result)
except (SyntaxError, ValueError):
    print("Invalid input")

Why use ast.literal_eval?

The input looks like a Python tuple: (20, 5, 5).

ast.literal_eval safely converts simple literal values such as tuples and integers. Do not use eval for user input because it can execute arbitrary Python code.

8. Trace the program

Input:

(20, 8, 3)

The program does this:

1. Reads T=20, M=8, B=3.
2. Checks 20 > 0: valid.
3. Checks 0 < 8 ≤ 20: valid.
4. Checks 0 ≤ 3 ≤ 8: valid.
5. Calculates B / M = 3 / 8 = 0.375.
6. Rounds to two decimal places: 0.38.

Output:

0.38

Formula summary

Basic probability:         P(M) = M / T
Intersection probability:  P(M ∩ S) = B / T
Conditional probability:   P(A | B) = P(A ∩ B) / P(B)
This question:             P(S | M) = (B / T) / (M / T) = B / M
Validation:                T > 0, 0 < M ≤ T, 0 ≤ B ≤ M

Practice

1. What should the program print for (50, 20, 15)?
2. Why is (30, 0, 0) invalid for this specific question?
3. Why is (10, 11, 2) invalid?

Answers

1. 15 / 20 = 0.75
2. It would require dividing by zero; there are no Math-passing students to condition on.
3. 11 Math-passing students cannot come from a class of only 10 students.