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Introduction to Probability - Estimate empirical probability using experimental and observed data

Grade 9CBSE

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

🔑Concepts

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Empirical probability, also known as experimental probability, is based on actual experiments and the recorded frequency of outcomes.

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A 'trial' is an action which results in one or several outcomes. For example, tossing a coin once or throwing a die once is a trial.

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An 'event' for an experiment is the collection of some outcomes of the experiment. We usually denote it by the letter EE.

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The probability of an event EE is a number such that 0≤P(E)≤10 \le P(E) \le 1.

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An event that is impossible has a probability of 00, and an event that is certain to occur has a probability of 11.

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The sum of the probabilities of all possible mutually exclusive events in an experiment is always 11.

📐Formulae

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

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

💡Examples

Problem 1:

A coin is tossed 500500 times with the following frequencies: Head: 245245, Tail: 255255. Compute the probability for each event.

Solution:

Total number of trials = 500500. Frequency of Head (E1E_1) = 245245. Frequency of Tail (E2E_2) = 255255. P(E1)=245500=0.49P(E_1) = \frac{245}{500} = 0.49 P(E2)=255500=0.51P(E_2) = \frac{255}{500} = 0.51

Explanation:

To find the empirical probability, we divide the frequency of the specific outcome by the total number of coin tosses. Note that 0.49+0.51=10.49 + 0.51 = 1.

Problem 2:

In a survey of 200200 students, it was found that 135135 like Mathematics while the rest dislike it. Find the probability that a student chosen at random dislikes Mathematics.

Solution:

Total number of students = 200200. Number of students who like Mathematics = 135135. Number of students who dislike Mathematics is calculated as: 200−13565\begin{array}{r} 200 \\ - 135 \\ \hline 65 \end{array} Let EE be the event that a student dislikes Mathematics. P(E)=65200=1340=0.325P(E) = \frac{65}{200} = \frac{13}{40} = 0.325

Explanation:

First, we subtract the number of students who like the subject from the total to find those who dislike it. Then, we apply the probability formula: favorable outcomestotal outcomes\frac{\text{favorable outcomes}}{\text{total outcomes}}.

Problem 3:

A die is thrown 10001000 times with the frequencies for the outcomes 1,2,3,4,51, 2, 3, 4, 5 and 66 as given in the following table: Outcome 11: 179179, Outcome 22: 150150, Outcome 33: 157157, Outcome 44: 149149, Outcome 55: 175175, Outcome 66: 190190. Find the probability of getting an outcome greater than 44.

Solution:

Total trials = 10001000. Outcomes greater than 44 are 55 and 66. Sum of frequencies of 55 and 66: 175+190365\begin{array}{r} 175 \\ + 190 \\ \hline 365 \end{array} Let EE be the event of getting an outcome >4> 4. P(E)=3651000=0.365P(E) = \frac{365}{1000} = 0.365

Explanation:

To find the probability of a compound event (getting 55 or 66), we add the frequencies of the individual favorable outcomes and divide by the total number of trials.