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Learning Resources

Probability

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 09 Pathway: N/A

First Principles

Objective: Calculate the probabilities of combined events (independent and dependent) using sample space grids, tree diagrams, and multiplication rules.

Interactive Combined Event Explorer

Simulate combined probabilities for independent or dependent draws!

(a) Concrete Scenario: Every morning in Nairobi, Wanjiku takes a matatu to the bus station, then boards a connecting bus to school. The matatu is on time \(80\%\) of the days (\(P(M) = 0.8\)), and the connecting bus is on time \(70\%\) of the days (\(P(B) = 0.7\)). What is the probability that both arrive on time on a given morning?

(b) Key Definitions:

  • Independent Events: The outcome of the first event does not affect the probability of the second event (e.g., rolling two dice).
  • Dependent Events: The outcome of the first event changes the sample space or probability of subsequent events (e.g., drawing cards or marbles without replacement).
  • Combined Probability Rule (AND): \(P(A \cap B) = P(A) \times P(B)\) for independent events.

(c) Sample Space Grid & Multiplication: For Wanjiku's commute: \[P(\text{Both On Time}) = P(M) \times P(B) = 0.8 \times 0.7 = 0.56 = 56\%\] For dependent events (e.g., drawing 2 red marbles without replacement from 5 red, 7 blue): \[P(R_1 \cap R_2) = P(R_1) \times P(R_2 \mid R_1) = \frac{5}{12} \times \frac{4}{11} = \frac{5}{33}\]

Rule of Thumb: "AND" means MULTIPLY probabilities along branches; "OR" means ADD probabilities across mutually exclusive outcomes!

Key Formulas

\(P(A \cap B) = P(A) \times P(B)\) — Probability of both independent events \(A\) AND \(B\) occurring.
\(P(A \cap B) = P(A) \times P(B \mid A)\) — Multiplication rule for dependent events (without replacement).
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) — Addition rule for finding the probability of event \(A\) OR event \(B\) occurring.
\(P(A') = 1 - P(A)\) — Complementary probability (probability that event \(A\) does NOT occur).
\(P(\text{At least one}) = 1 - P(\text{None})\) — Quick method for solving multi-trial non-occurrence problems.

Worked Examples

Example 1 (Easy - Independent Events): A fair coin is tossed and a standard 6-sided die is rolled. Find the probability of getting a Head on the coin AND a 4 on the die.
  1. Determine independence: The coin flip does not affect the die roll.
  2. Individual probabilities: \(P(H) = \frac{1}{2}\), \(P(4) = \frac{1}{6}\).
  3. Multiply: \(P(H \cap 4) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}\).
  4. Answer: \(\frac{1}{12}\).
Example 2 (Medium - Commute Reliability): A matatu is on time with probability \(0.8\) and a connecting train is on time with probability \(0.7\). Assuming independent events, calculate: (a) Probability both are on time, and (b) Probability at least one is on time.
  1. Both on time: \(P(M \cap T) = 0.8 \times 0.7 = 0.56\).
  2. At least one on time: \(P(M \cup T) = P(M) + P(T) - P(M \cap T) = 0.8 + 0.7 - 0.56 = 0.94\).
  3. Answer: (a) \(0.56\) ; (b) \(0.94\).
Example 3 (Hard - Dependent Draw Without Replacement): A box contains \(5\) red marbles, \(4\) blue marbles, and \(3\) green marbles (total \(12\)). Two marbles are drawn sequentially without replacement. Find the probability that both marbles are red.
  1. First draw: \(P(R_1) = \frac{5}{12}\).
  2. Second draw: One red marble is gone, leaving \(4\) red out of \(11\) total. \(P(R_2 \mid R_1) = \frac{4}{11}\).
  3. Multiply dependent probabilities: \(P(R_1 \cap R_2) = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}\).
  4. Answer: \(\frac{5}{33}\).

Common Mistakes

Mistake Adding probabilities instead of multiplying them when finding the chance of two independent events occurring together (e.g., computing \(0.8 + 0.7 = 1.5\)).
Correction Probabilities can never exceed \(1.0\)! "AND" operations require multiplication of fractional probabilities (\(0.8 \times 0.7 = 0.56\)).
Why it feels right Students confuse "combined" with arithmetic addition.
Mistake Keeping the total sample space fixed (e.g., dividing by 12 on the second draw) when items are taken without replacement.
Correction Without replacement means both the numerator and denominator decrease by 1 for the second trial!

Real World

Transport Network Planning: Transit operators calculate combined transfer connection reliability across multi-stage bus and train routes in Nairobi.
Agricultural Risk Assessment: Farmers evaluate the joint probability of adequate rainfall (\(P = 0.65\)) and pest resistance (\(P = 0.80\)) to predict successful harvest yields (\(0.65 \times 0.80 = 0.52\)).
Quality Control Diagnostics: Manufacturing plants use compound probabilities to ensure multi-stage electronics pass serial testing protocols.

Practice

A fair coin is flipped and a standard 6-sided die is rolled. What is the probability of getting a Tail on the coin AND an even number on the die? Express your answer as a simplified fraction. (Type only the fraction, e.g., 1/4)
Review the concepts above.
An M-Pesa agent serves three independent customers in a row. For each customer, the probability of depositing cash is 0.6. What is the probability that all three customers deposit cash? Give your answer to three decimal places. (Type only the number, e.g., 0.216)
Review the concepts above.
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles (total 10). If two marbles are drawn with replacement (the first marble is put back before drawing the second), what is the probability that both marbles drawn are blue? Express your answer as a fraction. (Type only the fraction, e.g., 9/100)
Review the concepts above.
A teacher in a Kenyan primary school has a box containing 5 red marbles, 4 blue marbles, and 3 green marbles (total 12). Two marbles are drawn from the box one after another without replacement. What is the probability that both marbles drawn are red? Express your answer as a simplified fraction. (Type only the fraction, e.g., 5/33)
Review the concepts above.
In a probability experiment, event J occurs with probability 0.48 and event K with probability 0.36. The two events occur together with probability 0.18. What is the conditional probability P(J | K)? Give your answer to two decimal places. (Type only the number, e.g., 0.50)
Review the concepts above.
A matatu route has two independent traffic lights. The probability of hitting a red light at the first junction is 0.4 and at the second junction is 0.5. What is the probability that the driver hits AT LEAST ONE red light? Give your answer as a decimal. (Type only the number, e.g., 0.7)
Review the concepts above.