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The section covers probability theory, distinguishing between theoretical and experimental probability. It explains key concepts such as equally likely outcomes, favorable outcomes, and provides examples of calculating probabilities in different scenarios, while emphasizing the definition and significance of events, including certain and impossible events.
Probability is a crucial component of both mathematics and its applications in various fields. This section delves into the theoretical approach to probability, using models like coin tossing and die rolling to explain key concepts. The definitions of probability are outlined:
\[ P(E) = \frac{\text{Number of favorable outcomes to event } E}{\text{Total number of possible outcomes}} \]
The section also emphasizes significant points such as the properties of certain and impossible events, and how understanding probability is essential for decision-making processes in various fields like finance, healthcare, and engineering.
Equally Likely Outcomes: The outcomes of an experiment where each has the same likelihood of occurring, e.g., flipping a fair coin or tossing a fair die.
Theoretical Probability: The probability calculated based on the assumption of equally likely outcomes, defined as:
Experimental Probability: Based on the actual outcomes from experiments.
Elementary Event: An event consisting of a single outcome.
Complementary Events: For any event E, the event 'not E' occurs when E does not occur, satisfying \[ P(E) + P(E') = 1 \].
When tossing a coin, it's rather neat, heads and tails are both discrete.
Imagine a fair coin toss where you can't predict the side. There's a 50% chance for each, no need to decide!
'C.E.S' stands for Certain, Elementary, Impossible. Help remembering types of events in probability.
Tossing a fair coin results in two equally likely outcomes: Heads or Tails, giving P(Heads) = 1/2.
Drawing a red ball from a bag containing 3 red and 5 blue balls gives P(Red) = 3/8.
In a fair six-sided die, rolling a number greater than 4 has favorable outcomes 5 and 6, so P(number > 4) = 2/6 = 1/3.
Term: Equally Likely Outcomes
Definition: Outcomes in an experiment that have the same chance of occurring.
Outcomes in an experiment that have the same chance of occurring.
Term: Theoretical Probability
Definition: Probability based on the assumption of equally likely outcomes.
Probability based on the assumption of equally likely outcomes.
Term: Experimental Probability
Definition: Probability derived from actual experiments and observations.
Probability derived from actual experiments and observations.
Term: Elementary Event
Definition: An event with a single outcome.
An event with a single outcome.
Term: Complementary Events
Definition: Events that are mutually exclusive; one event occurring means the other does not.
Events that are mutually exclusive; one event occurring means the other does not.
Term: Impossible Event
Definition: An event that cannot happen, with a probability of zero.
An event that cannot happen, with a probability of zero.
Term: Certain Event
Definition: An event that is guaranteed to happen, with a probability of one.
An event that is guaranteed to happen, with a probability of one.