Q.A die is thrown repeatedly until a six comes up. What is the sample space for this experiment?
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Sample Space Outcomes: The Complete Picture of "What Could Happen?"
Imagine you're about to flip a coin. Before it lands, you know two things are possible: heads or tails. That set of all possible results — {Heads, Tails} — is your sample space. Every single thing that could happen, listed out completely.
That's the core idea. Let's build from there.
The Intuition First
When you roll a six-sided die, you don't know which number will come up. But you do know the full list of possibilities: 1, 2, 3, 4, 5, or 6. That list is the sample space. Each individual possibility — like rolling a 4 — is an outcome (also called a sample point).
Think of it as the "universe" of your experiment. Nothing outside this set can happen. If you're drawing a card from a standard deck, the sample space is all 52 cards. If you're checking the weather tomorrow (sunny, rainy, cloudy), those three options form the sample space.
The sample space is exhaustive — it covers every possible result — and its outcomes are mutually exclusive (no two can happen at the same time in a single trial).
The Precise Statement
In probability theory, an experiment is any process with uncertain outcomes (rolling a die, drawing a card, measuring rainfall). The sample space, denoted by S or Ω, is the set of all possible outcomes of that experiment.
Each element of S is called an outcome (or sample point). For a fair coin toss:
S={H,T}
For a single die roll:
S={1,2,3,4,5,6}
For the number of heads in two coin tosses:
S={0,1,2}
S={all possible outcomes of the experiment}
Two Important Flavors
Discrete sample space — outcomes can be listed (countable). Coin tosses, die rolls, number of customers in a queue.
Continuous sample space — outcomes form a range (uncountable). The exact time a bus arrives (any real number between 0 and 60 minutes), the temperature at noon, the height of a randomly selected student.
For continuous cases, we write intervals: S={x∣0≤x≤60} for bus arrival time in minutes.
Why This Matters
Every probability question starts with the sample space. If you don't know what could happen, you can't calculate the chance of a specific event. The sample space is the foundation — get it right, and everything else follows. …
The sample space lists every possible run of non-six throws ending in a six. …
The sample space is infinite: it lists every possible sequence of non-six outcomes followed by the six that ends the experiment.
The die is thrown repeatedly until a six appears; the experiment can end on the 1st throw, 2nd throw, 3rd throw, and so on — there is no fixed upper bound, so the sample space is countably infinite.
- Ends on throw 1: outcome 6.
- Ends on throw 2: outcome (x,6) where x∈{1,2,3,4,5} — 5 outcomes.
- Ends on throw 3: outcome (x,y,6) where x,y∈{1,2,3,4,5} — 25 outcomes.
- ...and so on for every number of throws. …
- CBSE 2026Set ANNUAL1 markMCQQ.Number of elements in sample space when a coin is tossed three times will be:(a) 3(b) 4(c) 8(d) 2
›Reveal solutionSolution
For n independent tosses of a coin, the sample space has 2n equally likely outcomes.
Each toss has 2 outcomes, Head (H) or Tail (T). For 3 tosses, the total number of outcomes is 2×2×2=23=8.
…
- CBSE 2025Set ANNUAL1 markMCQQ.The number of elements in the sample space obtained in the throw of two coins simultaneously is(a) 2(b) 4(c) 8(d) 16
›Reveal solutionSolution
Two coins tossed together have 4 sample points.
Each coin independently has 2 outcomes (Head, Tail). By the multiplication principle, two coins together have 2×2=4 outcomes …
- CBSE 2025Set ANNUAL1 markMCQQ.The number of elements in the sample space obtained in the throw of two dice simultaneously is(a) 6(b) 12(c) 18(d) 36
›Reveal solutionSolution
Two dice thrown together have 36 sample points.
Each die has 6 faces (1,2,3,4,5,6). By the multiplication principle, two dice give …
- CBSE 2025Set ANNUAL1 markMCQQ.If an event has only one sample point in the sample space, then the event is called a/an:(a) Complementary event(b) Exhaustive event(c) Compound event(d) Simple event
›Reveal solutionSolution
By definition, an event with only one sample point in the sample space is called a simple or elementary event.
In probability theory, if an event E associated with a random experiment contains only a single outcome (sample point) of the sample space, it is called a simple event (also called an elementary even …
- CBSE 2025Set ANNUAL1 markQ.Write True or False: If an event has more than one sample point, it is called a compound event.
›Reveal solutionSolution
By definition, an event containing more than one sample point of the sample space is called a compound event.
In probability, events are classified based on the number of sample points they contain:
- An event with exactly one sample point is a simple (elementary) event. …
- CBSE 2024Set ANNUAL1 markQ.A coin is tossed thrice, then find the sample space.
›Reveal solutionSolution
The sample space for tossing a coin thrice has 8 outcomes: S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}.
Each toss independently results in Head (H) or Tail (T), giving 2 possibilities per toss. For 3 tosses, the total number of outcomes is 2×2×2=8. Listing all …
- CBSE 2024Set sz1 markMCQQ.Sample space of a coin when tossed twice is:(a) {H,H}(b) {HH,HT,TH,TT}(c) {HH,TT}(d) {TH,HT}
›Reveal solutionSolution
Tossing a coin twice gives the sample space {HH,HT,TH,TT}, 4 equally likely outcomes.
Each individual coin toss has 2 possible outcomes: H (Heads) or T (Tails). When the coin is tossed twice, the outcome is an ordered pair (first toss, second toss), so the total number of ou …
- CBSE 2024Set ANNUAL1 markMCQQ.If n coins are tossed simultaneously, what is the total number of outcomes?(a) 2n−1(b) 2n+1(c) 2n(d) 2n−2
›Reveal solutionSolution
Each coin independently lands Head or Tail (2 outcomes); by the multiplication principle, n coins give 2×2×⋯×2 (n times).
Each of the n coins has 2 possible outcomes (Head or Tail), and the coins are tossed independently.
…
- CBSE 2023Set ANNUAL1 markQ.Write down the sample space when a coin is tossed three times.
›Reveal solutionSolution
The sample space has 23=8 equally likely outcomes.
Each toss has 2 possible outcomes (H or T), and there are 3 tosses, so by the multiplication principle there are 23=8 outcomes: …
- CBSE 2021Set annual21 markQ.Define a simple event.
›Reveal solutionSolution
A simple event corresponds to a single outcome of the sample space.
For a random experiment with sample space S, an event is any subset of S. If an event E contains exactly one element of S (i.e., it corresponds to only one possible outcome of the experiment), it is called a simple event or elementary event.
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