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Probability - Random Experiments, Outcomes and Sample Spaces

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Random Experiment: An experiment is classified as random if its outcome cannot be predicted with absolute certainty, even when conducted under identical conditions. To be random, it must have more than one possible result. For example, tossing a coin is a random experiment because we cannot be sure if it will land on Heads or Tails before the toss occurs.

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Outcomes: A possible result of a random experiment is called an outcome. If you are drawing a single card from a standard deck, 'drawing the King of Hearts' is one specific outcome. Visually, an outcome can be thought of as a single point or a dot in a larger collection of possible results.

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Sample Space (SS): The set consisting of all possible outcomes of a random experiment is known as the sample space. It is usually denoted by the symbol SS. Visually, a sample space is represented as the universal set in a Venn diagram, often depicted as a rectangular box containing every possible outcome as a distinct point inside.

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Sample Point: Each individual element or outcome within a sample space is called a sample point. For example, if the sample space for tossing a coin twice is S={HH,HT,TH,TT}S = \{HH, HT, TH, TT\}, then each combination like HTHT is a sample point.

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Tree Diagrams: This is a visual representation used to list all possible outcomes of an experiment that occurs in multiple stages. It starts from a single node and splits into 'branches' for each possible outcome of the first stage; then, from the end of those branches, new branches split out for the next stage. This creates a path-like structure where each path from the start to a leaf represents a unique outcome in the sample space.

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Events as Subsets: An event is a specific collection of outcomes, which mathematically is a subset of the sample space SS. If we roll a die, the sample space is S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. An event EE of getting an even number would be E={2,4,6}E = \{2, 4, 6\}. Visually, an event is a region or a circle drawn inside the rectangular sample space.

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Mutually Exclusive Events: Two events AA and BB are said to be mutually exclusive if they cannot happen at the same time, meaning their intersection is an empty set (A∩B=∅A \cap B = \emptyset). Visually, in a Venn diagram, mutually exclusive events are shown as two separate circles that do not overlap or touch each other.

📐Formulae

Total number of outcomes for nn tosses of a coin: n(S)=2nn(S) = 2^n

Total number of outcomes for nn rolls of a die: n(S)=6nn(S) = 6^n

Theoretical probability of an event EE: P(E)=fracn(E)n(S)P(E) = \\frac{n(E)}{n(S)}

Range of probability: 0leP(E)le10 \\le P(E) \\le 1

Probability of an impossible event: P(emptyset)=0P(\\emptyset) = 0

Probability of a sure event: P(S)=1P(S) = 1

💡Examples

Problem 1:

A coin is tossed. If the outcome is a Head, a die is thrown. If the outcome is a Tail, the coin is tossed one more time. Describe the sample space for this experiment.

Solution:

Step 1: Identify the first stage outcomes. The first toss results in either Heads (HH) or Tails (TT). Step 2: Follow the branch for Heads. If HH occurs, we roll a die. The possible outcomes are {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}. This gives us outcomes: (H,1),(H,2),(H,3),(H,4),(H,5),(H,6)(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6). Step 3: Follow the branch for Tails. If TT occurs, we toss the coin again. The possible outcomes for the second toss are {H,T}\{H, T\}. This gives us outcomes: (T,H),(T,T)(T, H), (T, T). Step 4: Combine all paths to form the sample space. S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,H),(T,T)}S = \{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, H), (T, T)\}

Explanation:

The problem describes a sequential experiment. We use a branching logic (similar to a tree diagram) to ensure every possible path of the experiment is recorded. Each path from the first action to the final action constitutes one sample point.

Problem 2:

Two dice are thrown simultaneously. Find the sample space size n(S)n(S) and the probability that the sum of the numbers appearing on the dice is exactly 7.

Solution:

Step 1: Calculate the total number of outcomes. Since each die has 6 faces and there are two dice, n(S)=6times6=36n(S) = 6 \\times 6 = 36. Step 2: Define the event EE as 'the sum is 7'. List all pairs (x,y)(x, y) such that x+y=7x + y = 7: E={(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}E = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} Step 3: Count the number of favorable outcomes: n(E)=6n(E) = 6. Step 4: Calculate the probability using the formula: P(E)=fracn(E)n(S)=frac636=frac16P(E) = \\frac{n(E)}{n(S)} = \\frac{6}{36} = \\frac{1}{6}

Explanation:

For experiments involving two dice, we treat the dice as distinct (or thrown in sequence) to form ordered pairs. There are 36 possible pairs, and by systematically listing pairs that add up to 7, we find there are 6 such outcomes.