📚 Concepts

📚 Probability

• Probability is the measure of how likely an event is to occur. It lies between 0 and 1.

📚 Experiment

• An action or process that leads to one or several outcomes.

• E.g., tossing a coin or rolling a die.

📚 Trial

• A single performance of an experiment.

• E.g., tossing a coin once is one trial.

📚 Outcome

• The result of a single trial.

• E.g., “Head” when a coin is tossed.

📚 Sample Space (S)

• The set of all possible outcomes of an experiment.

• E.g., S = {1, 2, 3, 4, 5, 6} when a die is thrown.

  
  

📚 Event (E)

• A collection of one or more outcomes.

• E.g., Getting an even number on a die = {2, 4, 6}

📚 Favourable Outcomes

• The outcomes that satisfy the condition of the event.

• E.g., Favourable outcomes for getting a prime number on a die = {2, 3, 5}

  
  

📚 Equally Likely Outcomes

• When all outcomes of an experiment have an equal chance of occurring.

• E.g., Getting head or tail in a fair coin toss.

📚 Empirical Probability (Experimental Probability)

• The probability based on actual experiments or observations.

📚 Theoretical Probability

• The probability calculated based on reasoning, not actual experiments.

✍️ Formulas

✍️ Theoretical Probability of an Event𝐏(𝐄) = 𝐅𝐚𝐯𝐨𝐮𝐫𝐚𝐛𝐥𝐞 𝐎𝐮𝐭𝐜𝐨𝐦𝐞𝐬 ÷ 𝐓𝐨𝐭𝐚𝐥 𝐎𝐮𝐭𝐜𝐨𝐦𝐞𝐬

• 𝐏(𝐄) = Probability of the event E

• Favourable Outcomes = Outcomes where event E occurs

• Total Outcomes = Total number of equally likely outcomes

✍️ Experimental (Empirical) Probability𝐏(𝐄) = 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐓𝐢𝐦𝐞𝐬 𝐄𝐯𝐞𝐧𝐭 𝐎𝐜𝐜𝐮𝐫𝐬 ÷ 𝐓𝐨𝐭𝐚𝐥 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐓𝐫𝐢𝐚𝐥𝐬

• Trials = Number of times experiment is repeated

• Event Occurs = Count of times the desired outcome happened

✍️ Range of Probability

𝟎 ≤ 𝐏(𝐄) ≤ 𝟏

• 𝐏(𝐄) = 0 → Impossible event

• 𝐏(𝐄) = 1 → Certain event

✍️ Complementary Probability

𝐏 (𝐍𝐨𝐭 𝐄) = 𝟏 − 𝐏(𝐄)

• 𝐏 (𝐍𝐨𝐭 𝐄) = Probability that event E does not happen

✍️ Total Probability

𝐏(𝐄) + 𝐏 (𝐍𝐨𝐭 𝐄) = 𝟏

• Sum of probabilities of an event and its complement is always 1