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Introduction to Probability - Explain randomness and classify events by likelihood

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 numerical measure of the likelihood that a particular event will occur.

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A random experiment is an action where the result cannot be predicted with absolute certainty. The possible results are called outcomes.

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The Empirical (or Experimental) Probability of an event EE, denoted by P(E)P(E), is calculated based on the actual results of an experiment: P(E)=Number of trials in which the event happenedTotal number of trialsP(E) = \frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}}

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The value of probability always lies between 00 and 11, inclusive. This is expressed as 0≤P(E)≤10 \le P(E) \le 1.

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Events can be classified by likelihood: An Impossible Event has P(E)=0P(E) = 0; a Sure or Certain Event has P(E)=1P(E) = 1; and events with a 0.50.5 chance are called Equally Likely.

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The sum of the probabilities of all possible outcomes in an experiment is always equal to 11. If there are nn outcomes, then P(E1)+P(E2)+...+P(En)=1P(E_1) + P(E_2) + ... + P(E_n) = 1.

📐Formulae

P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}

0≤P(E)≤10 \le P(E) \le 1

P(Event)+P(Not Event)=1P(\text{Event}) + P(\text{Not Event}) = 1

P(Sure Event)=1P(\text{Sure Event}) = 1

P(Impossible Event)=0P(\text{Impossible Event}) = 0

💡Examples

Problem 1:

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

Solution:

P(Head)=4551000=0.455P(\text{Head}) = \frac{455}{1000} = 0.455 P(Tail)=5451000=0.545P(\text{Tail}) = \frac{545}{1000} = 0.545

Explanation:

To find the experimental probability, divide the frequency of the specific outcome by the total number of trials. Note that 0.455+0.545=10.455 + 0.545 = 1.

Problem 2:

A die is thrown 500500 times. The frequency of the outcome '66' is 8080. What is the probability of getting a '66'? Classify this likelihood.

Solution:

P(6)=80500=850=0.16P(6) = \frac{80}{500} = \frac{8}{50} = 0.16

Explanation:

Since the probability 0.160.16 is closer to 00 than to 11, the event is considered 'Unlikely' but not impossible.

Problem 3:

In a survey of 200200 students, 135135 like Statistics while 6565 do not like it. Find the probability that a student chosen at random does not like Statistics.

Solution:

Total number of students =200= 200. Number of students who do not like Statistics =65= 65. P(Not liking Statistics)=65200=1340=0.325P(\text{Not liking Statistics}) = \frac{65}{200} = \frac{13}{40} = 0.325

Explanation:

The probability is calculated by taking the count of the favorable event (students who do not like Statistics) and dividing by the total population surveyed.