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The Mathematics of Maybe: Introduction to Probability - Elements of Probability: Sample Spaces and Events

Grade 9CBSE

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

🔑Concepts

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Probability is the branch of mathematics that deals with the measurement of uncertainty of various phenomena.

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An Experiment is an operation which can produce some well-defined outcomes. Each performance of an experiment is called a Trial.

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The Sample Space (SS) is the set of all possible outcomes of an experiment. For example, if a coin is tossed, S={H,T}S = \{H, T\}.

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An Event (EE) is a collection of some outcomes of an experiment, which is a subset of the sample space.

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The Empirical (Experimental) Probability of an event is based on the actual results of an experiment and the number of trials performed.

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The probability of an event EE, denoted by P(E)P(E), always satisfies the condition: 0≤P(E)≤10 \le P(E) \le 1.

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An Impossible Event has a probability of 00, while a Sure Event has a probability of 11.

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The sum of the probabilities of all the elementary events of an experiment is 11.

📐Formulae

P(E)=Number of trials in which the event occurredTotal number of trialsP(E) = \frac{\text{Number of trials in which the event occurred}}{\text{Total number of trials}}

P(E)+P(not E)=1P(E) + P(\text{not } E) = 1

0≤P(E)≤10 \le P(E) \le 1

💡Examples

Problem 1:

A coin is tossed 10001000 times with the following frequencies: Head: 455455, Tail: 545545. Compute the probability for each event.

Solution:

Total number of trials = 10001000. Let E1E_1 be the event of getting a head and E2E_2 be the event of getting a tail. P(E1)=Number of headsTotal trials=4551000=0.455P(E_1) = \frac{\text{Number of heads}}{\text{Total trials}} = \frac{455}{1000} = 0.455 P(E2)=Number of tailsTotal trials=5451000=0.545P(E_2) = \frac{\text{Number of tails}}{\text{Total trials}} = \frac{545}{1000} = 0.545

Explanation:

To find the experimental probability, we divide the frequency of the specific outcome by the total number of trials conducted.

Problem 2:

A die is thrown 500500 times. The frequency of the outcome 33 is 7575. Find the probability of getting a 33.

Solution:

Total number of trials = 500500. Frequency of outcome 3=753 = 75. P(3)=Frequency of 3Total number of trials=75500P(3) = \frac{\text{Frequency of 3}}{\text{Total number of trials}} = \frac{75}{500} On simplifying: P(3)=320=0.15P(3) = \frac{3}{20} = 0.15

Explanation:

The probability is calculated by taking the ratio of the occurrences of the event (getting a 3) to the total number of times the die was thrown.

Problem 3:

In a survey of 200200 students, 135135 liked statistics while 6565 disliked it. Find the probability that a student chosen at random dislikes statistics.

Solution:

Total number of students = 200200. Number of students who dislike statistics = 6565. Let EE be the event that a student dislikes statistics. P(E)=65200P(E) = \frac{65}{200} P(E)=1340=0.325P(E) = \frac{13}{40} = 0.325

Explanation:

The probability is the number of students in the specific category (dislike) divided by the total sample size (surveyed students).