independent event

Examples of independent event in the following topics:

• Independent Events

  • Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs.
  • For example, the outcomes of two roles of a fair die are independent events.
  • To show two events are independent, you must show only one of the above conditions.
  • If two events are NOT independent, then we say that they are dependent.
  • The events are considered to be dependent or not independent.

• Independence

  • The concept of independence extends to dealing with collections of more than two events.
  • For independent events, the condition does not change the probability for the event.
  • Consider a fair die role, which provides another example of independent events.
  • First, note that each coin flip is an independent event.
  • Finally, the concept of independence extends to collections of more than 2 events.

• Practice 2: Calculating Probabilities

  • Students will determine whether two events are mutually exclusive or whether two events are independent.
  • Are L and C independent events?
  • Are L and C mutually exclusive events?

• The Multiplication Rule

  • As an example, suppose that we draw two cards out of a deck of cards and let $A$ be the event the the first card is an ace, and $B$ be the event that the second card is an ace, then:
  • Note that when $A$ and $B$ are independent, we have that $P(B|A)= P(B)$, so the formula becomes $P(A \cap B)=P(A)P(B)$, which we encountered in a previous section.
  • The probability that we get a 2 on the die and a tails on the coin is $\frac{1}{6}\cdot \frac{1}{2} = \frac{1}{12}$, since the two events are independent.

• Student Learning Outcomes

  • Determine whether two events are mutually exclusive and whether two events are independent.

• Independence

  • Similarly, if there are k events A1, ..., Ak from k independent processes, then the probability they all occur is P(A1) x P(A2) x ... x P(Ak)
  • The question we are asking is, are the occurrences of the two events independent?
  • We say that two events A and B are independent if they satisfy Equation (2.29).
  • If we shuffle up a deck of cards and draw one, is the event that the card is a heart independent of the event that the card is an ace?
  • P(H) x P(ace) =1/4 x 1/13 = 1/52 = P(H and ace)Because the equation holds, the event that the card is a heart and the event that the card is an ace are independent events.

• Mutually Exclusive Events

  • Are G and H independent?
  • For practice, show that P(H|G) = P(H) to show that G and H are independent events.
  • Are the events of being female and having long hair independent?
  • The events of being female and having long hair are not independent.
  • The events of being female and having long hair are not independent; knowing that a student is female changes the probability that a student has long hair.

• Conditional Probability

  • The conditional probability of an event is the probability that an event will occur given that another event has occurred.
  • Our estimation of the likelihood of an event can change if we know that some other event has occurred.
  • Suppose that $B$ is the event that at least one heads occurs and $A$ is the event that all 3 coins are the same.
  • The reason behind this is that the occurrence of event $A$ may provide extra information that can change the probability that event $B$ occurs.
  • If the knowledge that event $A$ occurs does not change the probability that event $B$ occurs, then $A$ and $B$ are independent events, and thus, $P(B|A) = P(B)$.

• Checking for independence

  • Such a low probability indicates a rare event.
  • The difference of 29.2% being a rare event suggests two possible interpretations of the results of the study:
  • H0: Independence model.
  • Errors do occur, just like rare events, and we might choose the wrong model.
  • A stacked dot plot of differences from 100 simulations produced under the independence model, H0, where gender sim and decision are independent.

• Fundamentals of Probability

  • An impossible event, or an event that never occurs, has a probability of 0.
  • Two events $A$ and $B$ are independent if knowing that one occurs does not change the probability that the other occurs.
  • If $A$ and $B$ are independent, then $P(A \ \text{and} \ B) = P(A)P(B)$.
  • Therefore when A and B are independent, we have $P(A \cap B) = P(A)P(B).$
  • If $A$ is the event that the first flip is a heads and $B$ is the event that the second flip is a heads, then $A$ and $B$ are independent.