Grade 10 Probability: Short Notes

10 ශ්‍රේණිය · 29 පරිච්ඡේදය · සම්භාවිතාව

74%

of Grade 10 term tests include it (40 of 54)

448

marks in past papers (67 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Find P(A) and P(A′) for simple and compound events
Tell whether two events are mutually exclusive
Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Use P(A ∩ B) = P(A) × P(B) for independent events
Show a two-stage experiment on a grid and on a tree diagram and find probabilities

A spin and a toss

At the school Avurudu pola, a stall asks you to toss a coin and spin a wheel numbered 1 to 6. You win a gift only if you get a head and a 6. Is it worth Rs 50 a try? This chapter shows how to work out chances like this exactly, using grids and tree diagrams.

Quick recap

A random experiment has outcomes we can't predict for sure.• Sample space S: the set of all possible outcomes.• Event A: a subset of S.If all outcomes are equally likely:P(A) = n(A) / n(S)0 ≤ P(A) ≤ 1. P = 0 means impossible; P = 1 means certain.

Simple, compound and complement

Rolling a die:• Simple event: just one outcome, e.g. getting a 4.• Compound event: two or more outcomes, e.g. getting an even number {2, 4, 6}.• Complement A′: everything in S that is not in A.Since A and A′ together fill S:P(A′) = 1 − P(A)

Mutually exclusive or not?

Two events are mutually exclusive if they can't happen together (A ∩ B = ∅).Rolling a die:• A = {2, 4, 6} (even), B = {1, 3} (odd and less than 5): no common outcome, so mutually exclusive.• A = {2, 4, 6}, C = {4, 5, 6} (greater than 3): 4 and 6 are in both, so not mutually exclusive.

🧮 The 'or' rule

Use it for A or B (or both). Subtracting P(A ∩ B) stops the overlap being counted twice. If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

✏️ Numbered cards

Cards numbered 1 to 20 are shuffled and one is drawn. A: a multiple of 3. B: a multiple of 5. Find P(A ∪ B).

1A = {3, 6, 9, 12, 15, 18}, so P(A) = 6/20.
2B = {5, 10, 15, 20}, so P(B) = 4/20.
3A ∩ B = {15}, so P(A ∩ B) = 1/20. Not mutually exclusive.
4P(A ∪ B) = 6/20 + 4/20 − 1/20 = 9/20

✅ 9/20

⚠️ 'Or' means the union

Two errors examiners keep reporting:1. For "a multiple of 3 or 5", listing only one set. 'Or' means the union: either one, or both.2. Adding 6/20 + 4/20 = 10/20 and counting 15 twice. Check for a common outcome first; if there is one, subtract P(A ∩ B).

Independent events

Two events are independent if one happening doesn't change the chance of the other.• Tossing a coin and rolling a die.• Drawing a ball, putting it back, then drawing again.For independent events:P(A ∩ B) = P(A) × P(B)The Avurudu stall: P(head and 6) = 1/2 × 1/6 = 1/12. Not great odds!

✏️ Sample space on a grid

Spinner A shows 1, 2, 3 or 4; spinner B shows 1, 2 or 3, all equally likely. Both are spun. Find the probability that the sum is 5.

1Mark every pair on a grid: A along, B up. 4 × 3 = 12 outcomes.
2Circle the sum-5 points: (2, 3), (3, 2), (4, 1).
3P(sum is 5) = 3/12 = 1/4

✅ 1/4

Tree diagrams

A tree shows a two-stage experiment.• Draw one set of branches per stage; write the probability on each branch.• Branches from one point add up to 1.• Along a path (A then B): multiply.• For several paths that fit the event: add the path results.

⚠️ Multiply along, add across

Two tree slips that cost marks:1. Adding along a path (2/5 + 2/5) instead of multiplying. 'And then' means multiply.2. Forgetting the second order. "One of each" is red-then-white and white-then-red, so add both paths.Also, every branch pair must add to 1: 2/5 + 3/5 = 1.

📌 Remember

• P(A) = n(A)/n(S) and P(A′) = 1 − P(A).• Mutually exclusive (can't happen together): P(A ∪ B) = P(A) + P(B).• Otherwise: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).• Independent: P(A ∩ B) = P(A) × P(B).• Grid: count points. Tree: multiply along a path, add the paths you need.

🧾 Quick revision sheet

P(A) when outcomes are equally likely	n(A) / n(S)
P(A′) in terms of P(A)	1 − P(A)
Mutually exclusive events	Can't happen together: A ∩ B = ∅, so P(A ∪ B) = P(A) + P(B).
P(A ∪ B) when A and B overlap	P(A) + P(B) − P(A ∩ B)
Independent events	One doesn't affect the other: P(A ∩ B) = P(A) × P(B).
Tree diagram: along a path you…	multiply. Then add the paths that fit the event.
A coin and a die: how many outcomes on the grid?	2 × 6 = 12