Random experiment and sample space
Before a cricket match the captain tosses a coin. Nobody can say which side will come up, but we know the only possible results are head or tail.
A random experiment has several possible results and we cannot tell which will occur. Each result is an outcome. The set of all possible outcomes is the sample space S, and n(S) is the number of outcomes.
Formula: n(S) = number of all possible outcomes
Rule 1: List the sample space — Write every outcome once.
- Easy: One coin: S = {H, T}, n(S) = 2.
- Medium: One die: S = {1, 2, 3, 4, 5, 6}, n(S) = 6.
- Exam-type: Two coins: S = {HH, HT, TH, TT}, n(S) = 4.
Rule 2: Count with the product rule — Multiply the outcomes of each stage.
- Easy: Two dice: 6 × 6 = 36 outcomes.
- Medium: Three coins: 2 × 2 × 2 = 8 outcomes.
- Exam-type: A coin and a die: 2 × 6 = 12 outcomes.
Order matters in a pair like (2, 5): it is different from (5, 2).
Writing HT and TH as one outcome when two coins are tossed.
Stages multiply.
Practice questions
- 1Write the sample space for tossing one coin.
- 2How many outcomes are there when two coins are tossed?
- 3How many outcomes are there when one die is rolled?
- 4How many outcomes are there when two dice are rolled?
- 5How many outcomes are there when three coins are tossed?
- 6A bag has 3 red and 2 blue balls. One ball is drawn. How many outcomes if every ball is counted separately?
- 7A coin is tossed and then a die is rolled. Find n(S).
- 8A coin is tossed until a head appears, but at most 3 times. List the outcomes.
Show answers (try them yourself first)
- 1{H, T}
- 24
- 36
- 436
- 58
- 65
- 712
- 8H, TH, TTH, TTT (4 outcomes)
Events
In a game, you win if the die shows an even number. "Even number" is not a single result; it is a group of results.
An event is any subset of the sample space. A simple event has one outcome. A compound event has more. The sure event is S itself, and the impossible event is the empty set.
Formula: Event E ⊆ S
Rule 1: Write the event as a set — Pick the outcomes that satisfy the condition.
- Easy: Die, even number: {2, 4, 6}.
- Medium: Die, number greater than 4: {5, 6}.
- Exam-type: Die, number 7: empty set (impossible event).
Rule 2: Count favourable outcomes — n(E) counts the outcomes in the event.
- Easy: Two coins, at least one head: 3 outcomes (HH, HT, TH).
- Medium: Two dice, sum 7: 6 outcomes.
- Exam-type: Die, number less than 7: all 6 outcomes (sure event).
Sure event: probability 1. Impossible event: probability 0.
Forgetting an outcome such as TH when listing events for two coins.
Event = chosen part of the sample space.
Practice questions
- 1Die: write the event "even number".
- 2Die: write the event "number greater than 4".
- 3Die: what kind of event is "number 7"?
- 4Die: what kind of event is "number less than 7"?
- 5Two coins: how many outcomes make "at least one head"?
- 6Two dice: how many outcomes make "sum is 7"?
- 7Two dice: how many outcomes make "sum is 9"?
- 8Three coins: how many outcomes make "exactly two heads"?
Show answers (try them yourself first)
- 1{2, 4, 6}
- 2{5, 6}
- 3Impossible event (empty set)
- 4Sure event (all of S)
- 53
- 66
- 74
- 83
Algebra of events
Rain and a holiday: "rain or holiday", "rain and holiday", "no rain" are new events made from old ones.
For events A and B: A ∪ B (A or B) is in at least one; A ∩ B (A and B) is in both; A′ (not A) is outside A; A − B (A but not B) is in A and not in B. Also n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Rule 1: Union, intersection — Die: A = even numbers, B = {1, 2, 3}.
- Easy: A ∪ B = {1, 2, 3, 4, 6}.
- Medium: A ∩ B = {2}.
- Exam-type: n(A ∪ B) = 3 + 3 − 1 = 5.
Rule 2: Complement, difference — Same A and B.
- Easy: A′ = {1, 3, 5}.
- Medium: A − B = {4, 6}.
- Exam-type: (A ∪ B)′ = {5}.
A′ together with A makes the whole sample space.
Confusing "A − B" with "B − A".
Or is union, and is intersection.
Practice questions
- 1Die, A = {2, 4, 6}, B = {1, 2, 3}. Find A ∪ B.
- 2Find A ∩ B for the same sets.
- 3Find A′ for the same set A.
- 4Find A − B for the same sets.
- 5Find B′ for B = {1, 2, 3}.
- 6Find (A ∪ B)′ for the same sets.
- 7Die: A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {2, 4, 6}. Find A ∩ B ∩ C.
- 8For the same A, B, C find (A ∪ B) ∩ C.
Show answers (try them yourself first)
- 1{1, 2, 3, 4, 6}
- 2{2}
- 3{1, 3, 5}
- 4{4, 6}
- 5{4, 5, 6}
- 6{5}
- 7{4}
- 8{2, 4, 6}
Mutually exclusive and exhaustive events
In a match, "win" and "lose" cannot both happen on the same day, and together they cover (almost) every result.
Events are mutually exclusive if they cannot occur together (A ∩ B is empty). Events are exhaustive if their union is the whole sample space. A and A′ are always both.
Formula: Mutually exclusive: A ∩ B = ∅
Rule 1: Check each — Test the common outcomes and the union.
- Easy: Even {2, 4, 6} and odd {1, 3, 5}: no common outcome, union is S; mutually exclusive and exhaustive.
- Medium: {1, 2} and {2, 3}: common outcome 2; not mutually exclusive.
- Exam-type: {2, 3, 5} (prime) and {4, 6} (composite): mutually exclusive, but 1 is missing, so not exhaustive.
Rule 2: Overlapping events — They can still cover S.
- Easy: Two coins: "at least one head" and "at least one tail" share HT and TH.
- Medium: Their union is all four outcomes, so they are exhaustive but not mutually exclusive.
- Exam-type: Three sets can be exhaustive without being mutually exclusive.
Mutually exclusive means no overlap; exhaustive means no gap.
Treating "not mutually exclusive" as "not possible".
No overlap, no gap.
Practice questions
- 1Die: are "even number" and "odd number" mutually exclusive?
- 2Die: are {1, 2} and {2, 3} mutually exclusive?
- 3Die: are {1, 2}, {3, 4}, {5, 6} mutually exclusive and exhaustive?
- 4Two coins: are "at least one head" and "at least one tail" mutually exclusive? Exhaustive?
- 5Die: prime numbers and composite numbers. Exhaustive?
- 6Are A and A′ always mutually exclusive and exhaustive?
- 7Die: A = {1, 2, 3}, B = {3, 4, 5}, C = {5, 6}. Exhaustive? Mutually exclusive?
- 8If A = {1, 2} and B = {5} are mutually exclusive, find n(A ∪ B).
Show answers (try them yourself first)
- 1Yes (and exhaustive)
- 2No (2 is common)
- 3Yes to both
- 4Not mutually exclusive; exhaustive
- 5No, 1 is neither
- 6Yes
- 7Exhaustive; not mutually exclusive
- 83