Probability

Habari Mwanafunzi! Let's Uncover the Secrets of Chance.

Ever wondered about the chances of rain on the day of the school sports day? Or the probability of your favourite team, maybe Gor Mahia or AFC Leopards, winning their next match? It might seem like guesswork, but there's a powerful branch of mathematics that helps us understand and calculate these chances. Welcome to the world of Probability! This isn't just about exams; it's a tool that helps engineers, doctors, and even business leaders make smart decisions every single day. Let's get started and turn uncertainty into a number!

Section 1: The Basic Lingo (The Language of Chance)

Before we can calculate anything, we need to speak the language of probability. Think of these as our building blocks.

• Experiment: Any action where the result is uncertain. For example, tossing a coin, rolling a die, or picking a prefect from a list of candidates.
• Outcome: A single possible result of an experiment. If you roll a die, one outcome is '4'. Another is '1'.
• Sample Space (S): This is the big one! It's the set of all possible outcomes of an experiment. We usually write it inside curly braces { }.
• Event (E): The specific outcome or group of outcomes we are interested in. For example, in a die roll, an event could be 'getting an even number'.

Let's see the sample space for rolling a standard six-sided die:

   +-------+
  /       /|
 /       / +
+-------+  |
| o   o |  |   The Sample Space (S) is:
|   o   |  +   {1, 2, 3, 4, 5, 6}
| o   o | /
+-------+

An Event (E) could be "rolling an even number".
The outcomes for this event are {2, 4, 6}.

Section 2: The Magic Formula of Probability

At its heart, calculating basic probability is surprisingly straightforward. The probability of an event (E) happening is a fraction or a ratio:

P(E) =  Number of Favourable Outcomes
       -----------------------------
       Total Number of Possible Outcomes

Probability is always a value between 0 and 1.

• A probability of 0 means the event is impossible (e.g., the probability of the sun rising from the West).
• A probability of 1 means the event is certain (e.g., the probability that the sun will rise tomorrow).

Kenyan Example: A small bag contains 10 mangoes from Makueni. 7 are ripe (R) and 3 are unripe (U). If you pick one mango at random without looking, what is the probability that you pick a ripe one?

Solution:

• The event (E) is picking a ripe mango.
• Number of favourable outcomes (ripe mangoes) = 7.
• Total number of possible outcomes (all mangoes) = 10.

P(Ripe) = (Number of Ripe Mangoes) / (Total Number of Mangoes)
P(Ripe) = 7 / 10
P(Ripe) = 0.7
    

So, there is a 0.7 (or 70%) chance of picking a ripe mango. Sawa?

Section 3: Types of Events - When Things Get Interesting

Not all events are simple. Let's look at how they can interact.

A. Mutually Exclusive Events

These are events that cannot happen at the same time. For example, when you toss a coin, you can get a Head OR a Tail, but not both in the same toss. For these events, we use the Addition Rule.

P(A or B) = P(A) + P(B)

Scenario: A school has buses going to two different routes: Route A (e.g., Buruburu) and Route B (e.g., Westlands). The probability of a student taking Route A is 0.4 and Route B is 0.5. What is the probability that a randomly chosen student takes either Route A or Route B?

Solution: A student cannot be on both buses at the same time, so the events are mutually exclusive.

P(A or B) = P(A) + P(B)
P(A or B) = 0.4 + 0.5
P(A or B) = 0.9
    

B. Independent Events

These are events where the outcome of one does not affect the outcome of the other. For example, flipping a coin and rolling a die. The coin's result has no impact on the die's result. For these, we use the Multiplication Rule.

P(A and B) = P(A) * P(B)

Scenario: The probability of Kenya Power having a planned outage in your area on a given day is 0.1. The probability of it raining on that same day is 0.3. Assuming these events are independent, what is the probability that there is a power outage AND it rains?

Solution:

P(Outage and Rain) = P(Outage) * P(Rain)
P(Outage and Rain) = 0.1 * 0.3
P(Outage and Rain) = 0.03
    

There is a 0.03 (or 3%) chance of both happening on the same day. Not very likely, but possible!

Section 4: Visualizing with Tree Diagrams

When you have a sequence of events, things can get complicated. A tree diagram is a fantastic visual tool to map out all possible outcomes and their probabilities.

Example: A student takes a two-part exam. The probability of passing Paper 1 (P1) is 0.8. If they pass P1, the probability of passing Paper 2 (P2) is 0.9. If they fail P1, the probability of passing P2 is only 0.4. Let's find the probability of passing both papers.

                  START
                   / \
                  /   \
                 /     \
    P(P1)=0.8   /       \   P(F1)=0.2  (Note: 1 - 0.8 = 0.2)
               /         \
            Pass P1      Fail P1
             / \           / \
            /   \         /   \
           /     \       /     \
P(P2|P1)=0.9 P(F2|P1)=0.1 P(P2|F1)=0.4 P(F2|F1)=0.6
         /         \     /         \
        /           \   /           \
     Pass P2       Fail P2 Pass P2       Fail P2

OUTCOMES:
Pass P1 and Pass P2  --> P(P1 and P2) = 0.8 * 0.9 = 0.72  <-- This is our answer!
Pass P1 and Fail P2  --> P(P1 and F2) = 0.8 * 0.1 = 0.08
Fail P1 and Pass P2  --> P(F1 and P2) = 0.2 * 0.4 = 0.08
Fail P1 and Fail P2  --> P(F1 and F2) = 0.2 * 0.6 = 0.12

(Check: 0.72 + 0.08 + 0.08 + 0.12 = 1.00. Perfect!)

The probability of passing both papers is found by multiplying along the branches: 0.72 or 72%.

Image Suggestion: A colorful and clear digital illustration of the tree diagram above. The student at the start could be a Kenyan student in uniform, looking determined. The paths could be glowing lines, leading to outcomes labeled 'Pass Both!', 'Try Harder', etc., with the calculated probabilities shown clearly at the end of each branch.

Section 5: Conditional Probability - The "Given That..." Clause

This is a more advanced idea that you'll use a lot in data science and engineering. It's the probability of an event happening, given that another event has already occurred.

The notation is P(A|B), which reads "The probability of A, given B."

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

Kenyan Context: Let's say in a group of STEM students, the probability that a student uses an Airtel line and is studying Engineering is 0.15. The overall probability that any student in the group uses an Airtel line is 0.40. If we pick a student and we know they use Airtel, what is the probability they are also an Engineering student?

Solution:

• Event A: The student is in Engineering.
• Event B: The student uses Airtel.
• We want to find P(Engineering | Airtel).
• We know P(Engineering and Airtel) = 0.15
• We know P(Airtel) = 0.40

P(E|A) = P(E and A) / P(A)
P(E|A) = 0.15 / 0.40
P(E|A) = 0.375
    

So, if you know a student uses Airtel, there's a 37.5% chance they are in the Engineering faculty!

Summary & Your Challenge!

Excellent work! We've journeyed from the basics of experiments and outcomes to the power of tree diagrams and conditional probability. You've seen that probability is a logical and essential tool for navigating a world full of uncertainty.

Keep practicing, and you'll see these patterns everywhere, from weather forecasts on KTN to medical trial results. You've got this!

Challenge Question: In a class of 40 students preparing for their KCSE, 25 students take Physics, 20 students take Chemistry, and 8 students take both Physics and Chemistry. What is the probability that a randomly selected student takes at least one of the two subjects (i.e., Physics or Chemistry)?
Hint: These events are NOT mutually exclusive because a student can take both! You'll need the General Addition Rule: P(A or B) = P(A) + P(B) - P(A and B).