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The Mathematics of Maybe: Introduction to Probability - Measuring Probability Objectively

Grade 9CBSE

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

🔑Concepts

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Probability is a numerical way of expressing the degree of uncertainty of an event occurring.

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A trial is an action which results in one or several outcomes. For example, each toss of a coin or every throw of a die is a trial.

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An event (EE) for an experiment is the collection of some outcomes of the experiment.

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The Empirical (Experimental) Probability of an event is based on the actual results of an experiment and is calculated by the ratio of the frequency of the event to the total number of trials.

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The probability of any event EE always lies between 00 and 11 (inclusive), denoted as 0≤P(E)≤10 \le P(E) \le 1.

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An Impossible Event has a probability of 00. This occurs when the event cannot happen under the given conditions.

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A Sure Event (or Certain Event) has a probability of 11. This occurs when the event is guaranteed to happen.

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

📐Formulae

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}}

0≤P(E)≤10 \le P(E) \le 1

∑P(Ei)=P(E1)+P(E2)+...+P(En)=1\sum P(E_i) = P(E_1) + P(E_2) + ... + P(E_n) = 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. Frequency of Head (E1E_1) = 455455. Frequency of Tail (E2E_2) = 545545. P(E1)=4551000=0.455P(E_1) = \frac{455}{1000} = 0.455 P(E2)=5451000=0.545P(E_2) = \frac{545}{1000} = 0.545

Explanation:

To find the empirical probability, we divide the number of times the specific outcome (Head or Tail) occurred by the total number of coin tosses. Note that 0.455+0.545=10.455 + 0.545 = 1.

Problem 2:

A die is thrown 500500 times and the frequency of outcomes is recorded as follows: Outcome 11: 8080 times, Outcome 22: 7575 times, Outcome 33: 9090 times, Outcome 44: 7575 times, Outcome 55: 8585 times, Outcome 66: 9595 times. Find the probability of getting a number greater than 44.

Solution:

Total trials = 500500. An outcome greater than 44 means getting either 55 or 66. Number of times 55 or 66 appeared = 85+95=18085 + 95 = 180. Let EE be the event of getting a number greater than 44. P(E)=180500=1850=0.36P(E) = \frac{180}{500} = \frac{18}{50} = 0.36

Explanation:

The event consists of two favorable outcomes (55 and 66). We add their individual frequencies to get the total number of successful trials and divide by the total number of throws.

Problem 3:

In a survey of 200200 students, it was found that 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 = 200200. Number of students who do not like Statistics = 6565. Let EE be the event that a student does not like Statistics. P(E)=65200=1340=0.325P(E) = \frac{65}{200} = \frac{13}{40} = 0.325

Explanation:

The probability is calculated by taking the number of students who do not like the subject and dividing it by the total sample size of the survey.