Day 10: Understanding Probability: From Basics to Conditional

A Deep Dive into Probability with Real-Life Examples

What is Probability?

Basic Definition

Probability is the likelihood or chance of an event occurring. It's the ratio of:

Probability = Favorable outcomes / Total possible outcomes

Probability always ranges from 0 (impossible) to 1 (certain)

Probability Scale

0%	25%	50%	75%	100%
Impossible	Unlikely	Equal chance	Likely	Certain

Simple Examples:

Coin Toss:
- Getting heads: 1/2 = 0.5 = 50%
- Getting tails: 1/2 = 0.5 = 50%
Playing Cards:
- Drawing an ace: 4/52 = 1/13 ≈ 7.7%
- Drawing a heart: 13/52 = 1/4 = 25%

Understanding Conditional Probability

What Makes it "Conditional"?

Conditional probability is the likelihood of an event occurring, given that another event has already occurred.

P(A|B) = P(A and B) / P(B)

Read as: "Probability of A, given that B has occurred"

Real-Life Example: Weather and Traffic

Let's break down why traffic is more likely on rainy days:

Scenario	Probability
Chance of rain any day	30% = 0.3
Traffic on a normal day	80% = 0.8
Traffic when it rains	100% = 1.0

Why This Matters:

Normal traffic probability: 80%
But if it's raining (condition), traffic probability jumps to 100%
The rain condition changes the probability

Another Example: Biryani Orders

Understanding delivery success based on conditions:

Condition	Regular Day	Rainy Day
Delivery on time	90%	70%
Food hot on arrival	95%	80%

The weather condition affects both delivery time and food temperature!

Key Points to Remember

Basic Probability:
- Always between 0 and 1 (or 0% to 100%)
- All possible outcomes must sum to 1 (100%)
- Independent events don't affect each other
Conditional Probability:
- One event depends on another
- Changes based on known conditions
- Helps predict real-world scenarios better

Practice Problem Set: The Food Delivery Challenge

Scenario Setup

You run a food delivery service in Hitech City with the following data:

Total Orders per day: 100
Biryani orders: 40%
Peak hour orders (6-8 PM): 50%
Biryani orders during peak hours: 25 orders

Question 1: Basic Probability

What's the probability of a random order being a biryani order?

Solution:

Total orders = 100
Biryani orders = 40
Probability = 40/100 = 0.4 or 40%

Question 2: Conditional Probability

If an order comes during peak hours, what's the probability it's a biryani order?

Solution:

Peak hour orders = 50
Biryani orders during peak hours = 25
P(Biryani|Peak) = 25/50 = 0.5 or 50%

Question 3: Complex Scenario

Given these additional conditions:

Delivery success rate on normal days: 95%
Rainy days in a month: 6 days
Delivery success rate on rainy days: 80%

What's the probability of a successful delivery on a rainy day for a peak hour biryani order?

Step-by-Step Solution:

First, identify the events:
- A: Successful delivery
- B: Rainy day
- C: Peak hour biryani order
We know:
- P(A|B) = 80% = 0.8 (success rate on rainy days)
- P(C|B) = 50% = 0.5 (peak hour biryani probability)
Final Probability = 0.8 × 0.5 = 0.4 or 40%

Bonus Challenge: The Biryani Piece Probability

At Shah Gouse:

Each biryani has 3 pieces
60% are leg pieces
30% are breast pieces
10% are wing pieces

If you order 2 biryanis, what's the probability of getting at least one leg piece?

Solution:

First, find probability of NOT getting a leg piece in one biryani
- P(no leg) = 0.4
Probability of no leg pieces in both biryanis
- P(no leg in both) = 0.4 × 0.4 = 0.16
Therefore, probability of at least one leg piece
- P(at least one leg) = 1 - 0.16 = 0.84 or 84%

Key Learning Points

Basic probability deals with independent events
Conditional probability considers related events
Real-world scenarios often involve multiple conditions
Breaking down complex problems into simple steps helps

Day 10: Understanding Probability: From Basics to Conditional

What is Probability?

Basic Definition

Probability Scale

Simple Examples:

Understanding Conditional Probability

What Makes it "Conditional"?

Real-Life Example: Weather and Traffic

Why This Matters:

Another Example: Biryani Orders

Key Points to Remember

Practice Problem Set: The Food Delivery Challenge

Scenario Setup

Question 1: Basic Probability

Question 2: Conditional Probability

Question 3: Complex Scenario

Bonus Challenge: The Biryani Piece Probability

Key Learning Points

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