BONUS/REVIEW POWERPOINTS

PROVIDED BY MS. MERRELL

INDEPENDENT AND DEPENDENT EVENTS

Slide 2

INDEPENDENT EVENTS

WHATEVER HAPPENS IN ONE EVENT HAS ABSOLUTELY NOTHING TO DO WITH WHAT WILL HAPPEN NEXT BECAUSE:

1. THE TWO EVENTS ARE UNRELATEDOR

2. YOU REPEAT AN EVENT WITH AN ITEM WHOSE NUMBERS WILL NOT CHANGE (EG.: SPINNERS OR DICE)

OR3. YOU REPEAT THE SAME ACTIVITY, BUT YOU REPLACE THE

ITEM THAT WAS REMOVED.

The probability of two independent events, A and B, is equal to the probability of event A times the probability of event B.

P(A, B) = P(A) P(B)Slide 3

1

2

1

51 1 1

2 5 10

S

T

R

O

P1

2

3

6

5

4

Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel?

P(even) = (3 evens out of 6 outcomes)

(1 vowel out of 5 outcomes)P(vowel) =

P(even, vowel) =

Independent Events

Slide 4

DEPENDENT EVENT

• WHAT HAPPENS THE DURING THE SECOND EVENT DEPENDS UPON WHAT HAPPENED BEFORE.

• IN OTHER WORDS, THE RESULT OF THE SECOND EVENT WILL CHANGE BECAUSE OF WHAT HAPPENED FIRST.

The probability of two dependent events, A and B, is equal to the probability of event A times the probability of event B. However, the probability of event B now depends on event A.

P(A, B) = P(A) P(B)Slide 5

Dependent Event

6 3 or

14 7

5

13

3 5 15 or

7 13 91

Example: There are 6 black pens and 8 blue pens in a jar. If you take a pen without looking and then take another pen without replacing the first, what is the probability that you will get 2 black pens?

P(black second) = (There are 13 pens left and 5 are black)

P(black first) =

P(black, black) =

THEREFORE………………………………………………

Slide 6

TEST YOURSELFARE THESE DEPENDENT OR INDEPENDENT EVENTS?1. Tossing two dice and getting a 6 on both of them.

2. You have a bag of marbles: 3 blue, 5 white, and 12 red. You choose one marble out of the bag, look at it then put it back. Then you choose another marble.

3. You have a basket of socks. You need to find the probability of pulling out a black sock and its matching black sock without putting the first sock back.

4. You pick the letter Q from a bag containing all the letters of the alphabet. You do not put the Q back in the bag before you pick another tile.

Slide 7

FIND THE PROBABILITY

• P(JACK, FACTOR OF 12)

1

5

5

8x =5

40

1

8

Independent Events

Slide 8

FIND THE PROBABILITY

• P(6, NOT 5)1

6

5

6x =

5

36

Independent Events

Slide 9

FIND THE PROBABILITY

• P(Q, Q)

• ALL THE LETTERS OF THE ALPHABET ARE IN THE BAG 1 TIME

• DO NOT REPLACE THE LETTER

1

26

0

25x =

0

650

0

Dependent Events

Slide 10

CONDITIONAL PROBABILITY

• CONDITIONAL PROBABILITY IS THE PROBABILITY OF SOMETHING HAPPENING GIVEN A LIMITING CONDITION.

• WHAT IS THE PROBABILITY OF DRAWING A QUEEN OF SPADES GIVEN THE CARD IS BLACK?

• WHAT IS THE PROBABILITY OF ROLLING A 12 SIDED DIE AND GETTING A 4 GIVEN THE NUMBER IS EVEN?

1

26 /A

P A BB

1

6 /A

P A BB

FIND A CONDITIONAL PROBABILITY

THE TABLE SHOWS THE NUMBERS OF TROPICAL CYCLONES THAT FORMED DURING THE HURRICANE SEASONS FROM1988 TO 2004. USE THE TABLE TO ESTIMATE A) THE PROBABILITY THAT A FUTURE TROPICAL CYCLONE IS A HURRICANE AND B) THE PROBABILITY THAT A FUTURE TROPICAL CYCLONE IN THE NORTHERN HEMISPHERE IS A HURRICANE.

Type of Cyclone

Northern Hemisphere

Southern Hemisphere

Tropical depression

199 18

Tropical storm 398 200

Hurricane 545 215

DEPENDENT EVENTS

• IF A AND B ARE DEPENDENT EVENTS, THEN THE PROBABILITY THAT BOTH A AND B OCCUR IS:

P(B|A) IS READ “THE PROBABILITY THAT B OCCURS GIVEN THAT A HAS OCCURRED”

P A B P A P B A and

COMPARING INDEPENDENT AND DEPENDENT EVENTS

YOU RANDOMLY SELECT TWO CARDS FROM A STANDARD DECK OF 52 CARDS. WHAT IS THE PROBABILITY THAT THE FIRST CARD IS NOT A HEART AND THE SECOND IS A HEART IF

a) YOU REPLACE THE FIRST CARD BEFORE SELECTING THE SECOND

b) YOU DO NOT REPLACE THE FIRST CARD?Independent

Dependent

Use the information in the Cyclone table to find a)The probability that a future tropical cyclone is a tropical storm.

b) The probability that a future tropical cyclone in the Southern Hemisphere is a tropical storm.

Find the probability of drawing the given cards from a standard deck a) with replacement and b) without replacement.

A spade, then a club A jack, then another jack

• KEY CONCEPTS

• THE CONDITIONAL PROBABILITY OF B GIVEN A IS THE PROBABILITY THAT EVENT B OCCURS, GIVEN THAT EVENT A HAS ALREADY OCCURRED.

• IF A AND B ARE TWO EVENTS FROM A SAMPLE SPACE WITH P(A) ≠ 0, THEN THE CONDITIONAL PROBABILITY OF B GIVEN A, DENOTED , HAS TWO EQUIVALENT EXPRESSIONS:

• THE SECOND FORMULA CAN BE REWRITTEN AS ,IS READ “THE PROBABILITY OF B GIVEN A.”

• USING SET NOTATION, CONDITIONAL PROBABILITY IS WRITTEN LIKE THIS:

• THE “CONDITIONAL PROBABILITY OF B GIVEN A” ONLY HAS MEANING IF EVENT A HAS OCCURRED. THAT IS WHY THE FORMULA FOR HAS THE REQUIREMENT THAT P(A)≠0.

• THE CONDITIONAL PROBABILITY FORMULA CAN BE SOLVED TO OBTAIN A FORMULA FOR P(A AND B), AS SHOWN ON THE NEXT SLIDE.

• KEY CONCEPTS, CONTINUED

• REMEMBER THAT INDEPENDENT EVENTS ARE TWO EVENTS SUCH THAT THE PROBABILITY OF BOTH EVENTS OCCURRING IS EQUAL TO THE PRODUCT OF THE INDIVIDUAL PROBABILITIES. TWO EVENTS A AND B ARE INDEPENDENT IF AND ONLY IF

• P(A AND B) = P(A) • P(B). USING SET NOTATION, . • THE OCCURRENCE OR NON-OCCURRENCE OF ONE EVENT HAS NO EFFECT ON THE

PROBABILITY OF THE OTHER EVENT.

• IF A AND B ARE INDEPENDENT, THEN THE FORMULA FOR P(A AND B) IS THE EQUATION USED IN THE DEFINITION OF INDEPENDENT EVENTS, AS SHOWN BELOW.

Write the conditional probability formula.

Multiply both sides by P(A).

Simplify.

Reverse the left and right sides.

formula for P(A and B)

formula for P(A and B) if A and B are independent

• KEY CONCEPTS, CONTINUED

• THE FOLLOWING STATEMENTS ARE EQUIVALENT. IN OTHER WORDS, IF ANY ONE OF THEM IS TRUE, THEN THE OTHERS ARE ALL TRUE.

• EVENTS A AND B ARE INDEPENDENT.

• THE OCCURRENCE OF A HAS NO EFFECT ON THE PROBABILITY OF B; THAT IS,

• THE OCCURRENCE OF B HAS NO EFFECT ON THE PROBABILITY OF A; THAT IS,

• P(A AND B) = P(A) • P(B).

• NOTE: FOR REAL-WORLD DATA, THESE MODIFIED TESTS FOR INDEPENDENCE ARE SOMETIMES USED:

• EVENTS A AND B ARE INDEPENDENT IF THE OCCURRENCE OF A HAS NO SIGNIFICANT EFFECT ON THE PROBABILITY OF B; THAT IS,

• EVENTS A AND B ARE INDEPENDENT IF THE OCCURRENCE OF B HAS NO SIGNIFICANT EFFECT ON THE PROBABILITY OF A; THAT IS,

• WHEN USING THESE MODIFIED TESTS, GOOD JUDGMENT MUST BE USED WHEN DECIDING WHETHER THE PROBABILITIES ARE CLOSE ENOUGH TO CONCLUDE THAT THE EVENTS ARE INDEPENDENT.