Probability Experiments

Probability Experiments Basics

• A probability experiment is a chance process or situation that leads to at least two different outcomes.
• The set of all possible results from a probability experiment is known as the sample space. For instance, tossing a coin has a sample space of {Heads, Tails}.
• An event is an outcome or combination of outcomes from a probability experiment.

Probability Measure

• The probability of an event is measured on a scale from 0 to 1, where 0 indicates the event will not happen and 1 indicates the event is certain to happen.
• The probability of an event E, denoted as P(E), is calculated as the number of ways event E can happen divided by the total number of outcomes in the sample space.
• Complementary events are events that together make up the whole sample space. The probability of the complementary event E’, is P(E’) = 1 - P(E).

Independent and Dependent Events

• Two events are said to be independent if the occurrence of one event does not affect the probability of the occurrence of the other. For instance, tossing a coin and rolling a dice are independent events.
• If the probability of an event is influenced by whether another event occurs, the two events are said to be dependent. In this case, we often need to calculate conditional probability.

Conditional Probability

• Conditional probability is the probability of an event given that another event has already occurred.

• The conditional probability of an event A given that an event B has already occurred is denoted as P(A

  B) and calculated by the formula P(A and B) / P(B).

Addition Rule and Multiplication Rule

• The addition rule is used to calculate the probability that A or B happens for any two events A and B, defined as: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

• The multiplication rule is used to calculate the probability that both of two events A and B happen, defined as: P(A ∩ B) = P(A) * P(B

  A) for dependent events, or P(A) * P(B) for independent events.