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Impact of Past Events on Probabilities: Conditional Probability & Independence, Exams of Statistics

Western Carolina University (WCU)Statistics

The concepts of conditional probability and independence in the context of applied statistics. It defines conditional probability and provides a theorem for calculating it. The document also introduces the concepts of disjoint and independent events, and provides theorems and examples to illustrate these concepts. Students will learn how to use cross-tables or joint probability tables to find probabilities and test for independence.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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MATH 170 - Applied Statistics
Section 6.4 & 6.5: Key Ideas on Conditional Probability and Independence
Sometimes having extra information changes how you see things ... it’s the same way with proba-
bility. Sometimes, information on the occurrence of a previous event will effect the probability that
a related event will occur.
Definition 1 (Conditional Probability)
We denote the probability of the event Egiven that the event Fhas occurred by
P(E|F)
and call this a conditional probability.
Theorem 1 (Calculating Conditional Probability)
Let Eand Fbe two events with P(F)>0. The conditional probability of Egiven Foccurred,
P(E|F)is given by
P(E|F) = P(Eand F)
P(F)
We often use cross-tables or joint probability tables to answer questions about probabilities
(P(E), P(F), P(Eand F), P(E|F), etc). In cross-tables, we record frequencies, where as in joint
probabilities tables we record relative frequencies.
Example 1 Cross-Tables A large bowl has 120 candies in it. The candies are categorized according
to the cross-table below:
chocolate peppermint
hard 10 60
soft 40 10
A candy is selected at random from the bowl. Find the fol lowing probabilities:
1. P(the candy is chocolate) =
2. P(the candy is soft) =
3. P(the candy is NOT peppermint) =
4. P(the candy is both hard AND peppermint) =
5. P(the candy is either hard OR peppermint) =
6. P(the candy is hard, GIVEN that it is peppermint) = P(hard |peppermint) =
7. P(the candy is peppermint, GIVEN that it is hard) = P(peppermint |hard) =
pf2

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Download Impact of Past Events on Probabilities: Conditional Probability & Independence and more Exams Statistics in PDF only on Docsity!

MATH 170 - Applied Statistics Section 6.4 & 6.5: Key Ideas on Conditional Probability and Independence

Sometimes having extra information changes how you see things ... it’s the same way with proba- bility. Sometimes, information on the occurrence of a previous event will effect the probability that a related event will occur.

Definition 1 (Conditional Probability) We denote the probability of the event E given that the event F has occurred by

P (E|F )

and call this a conditional probability.

Theorem 1 (Calculating Conditional Probability) Let E and F be two events with P (F ) > 0. The conditional probability of E given F occurred, P (E|F ) is given by

P (E|F ) =

P (E and F ) P (F )

We often use cross-tables or joint probability tables to answer questions about probabilities (P (E), P (F ), P (E and F ), P (E|F ), etc). In cross-tables, we record frequencies, where as in joint probabilities tables we record relative frequencies.

Example 1 Cross-Tables A large bowl has 120 candies in it. The candies are categorized according to the cross-table below:

chocolate peppermint hard 10 60 soft 40 10

A candy is selected at random from the bowl. Find the following probabilities:

  1. P(the candy is chocolate) =
  2. P(the candy is soft) =
  3. P(the candy is NOT peppermint) =
  4. P(the candy is both hard AND peppermint) =
  5. P(the candy is either hard OR peppermint) =
  6. P(the candy is hard, GIVEN that it is peppermint) = P(hard | peppermint) =
  7. P(the candy is peppermint, GIVEN that it is hard) = P(peppermint | hard) =

Definition 2 (Disjoint Events) Two events E and F are said to be disjoint if they have no simple events in common. In terms of Venn Diagrams E ∩ F is empty.

Definition 3 (Independent Events) Two events E and F are said to be independent if the occurrence of one has no bearing on the probability of the other occurring, and in terms of formal probabilities: P (E|F ) = P (E)

Since P (E|F ) =

P (E and F ) P (F )

, we see from the definition of independent events:

P (E|F ) = P (E)

P (E and F ) P (F )

= P (E)

P (E and F ) = P (E) · P (F )

Theorem 2 (Test for Independence) Events E and F are independent if and only if

P (E and F ) = P (E) · P (F )

And since P (E|F ) = P^ (E P^ and (F ) F^ )even if E and F aren’t independent, we can multiply both sides

of this equation by P (F ) to get:

Theorem 3 (General Multiplication Rule for Two Events)

P (E and F ) = P (E|F ) · P (F )

Example 2 Sixty plastic balls are in a pen. They are categorized according to the cross-table below.

large small red 10 10 white 20 20

If a ball is randomly selected, find the following probabilities:

  1. P(the ball is red) =
  2. P(the ball is red GIVEN that it is large) = P(red | large) =
  3. Are the events “the ball is red” and “the ball is large” independent? Explain.

Homework:

  • pp. 302 - 305: # 29, 32, 33, 36, 37
  • pp. 312 - 315: # 41, 49, 53