# Ex: Determine the Probability of the Union of Two Events (OR)

On December 24, 2019 by Raul Dinwiddie

– A GROUP OF STUDENTS

TOOK A TEST. THE GRADES AND GENDER

ARE SUMMARIZED BELOW. IF ONE STUDENT IS CHOSEN

AT RANDOM FIND THE PROBABILITY

THAT THE STUDENT WAS FEMALE OR GOT A C ON THE TEST. SO WE WANT TO FIND

THE PROBABILITY THAT IF WE SELECT ONE STUDENT

IT WOULD BE A FEMALE OR THE STUDENT WOULD HAVE

A C ON THE TEST. THESE TWO EVENTS CAN OCCUR

AT THE SAME TIME SINCE THERE ARE SOME FEMALES THAT ALSO RECEIVED A C ON

THE TEST, AND THEREFORE THESE EVENTS ARE

NOT MUTUALLY EXCLUSIVE, AND THEREFORE, WE’LL HAVE TO APPLY THIS

PROBABILITY FORMULA HERE FOR THE PROBABILITY

OF THE UNION OF TWO EVENTS WHEN THEY ARE

NOT MUTUALLY EXCLUSIVE. THIS PROBABILITY IS EQUAL TO THE PROBABILITY

OF SELECTING A FEMALE + THE PROBABILITY

OF A STUDENT RECEIVING A C – THE PROBABILITY

OF A STUDENT BEING A FEMALE AND RECEIVING A C. WE’LL BE ABLE TO TELL

FROM THE TABLE WHY YOU HAVE TO SUBTRACT

THIS PROBABILITY HERE. NOTICE FROM THE TABLE,

LOOKING AT THE TOTAL, 65 STUDENTS TOOK THE TEST

AND NOTICE THIS ROW HERE GIVES US INFORMATION ABOUT

THE FEMALES. NOTICE HOW THERE WERE A TOTAL

OF 25 FEMALES THAT TOOK THE TEST AND THEREFORE THE PROBABILITY OF SELECTING

A FEMALE FROM THIS GROUP WOULD BE EQUAL

TO 25 DIVIDED BY 65 +– NOW WE WANT THE PROBABILITY THAT

A STUDENT SELECTED AT RANDOM WOULD HAVE A C ON THE TEST. THAT INFORMATION IS GIVEN

IN THIS COLUMN HERE. NOTICE HOW 12 OF THE STUDENTS

IN THE CLASS RECEIVED A C. AND THEREFORE, THE PROBABILITY OF SELECTING

A STUDENT THAT RECEIVED A C WOULD BE 12/65

OR 12 DIVIDED BY 65. BUT NOW, NOTICE HOW WE COUNTED

THESE 9 FEMALE STUDENTS THAT ALSO RECEIVED A C, THESE STUDENTS HERE

IN BOTH PROBABILITIES AND THAT’S THE REASON WHY WE

HAVE TO SUBTRACT THE PROBABILITY OF A STUDENT

BEING A FEMALE AND A C. WE CAN’T COUNT THEM TWICE TO GET

THE CORRECT PROBABILITY. SO THE PROBABILITY OF A STUDENT

BEING A FEMALE AND RECEIVING A C WOULD BE EQUAL

TO 9 DIVIDED BY 65. SO THE PROBABILITY OF ONE

STUDENT BEING SELECTED AND THE STUDENT BEING A FEMALE OR RECEIVING A C WOULD BE EQUAL

TO 25/65 + 12/65 – 9/65. NOTICE HOW WE HAVE

A COMMON DENOMINATOR OF 65 AND THE NUMERATOR IS 25 + 12

WHICH IS 37 – 9 WHICH IS 28. WHICH DOES NOT SIMPLIFY SO THIS

WOULD BE OUR EXACT PROBABILITY. LET’S ALSO EXPRESS THIS AS

A DECIMAL AND PERCENTAGE. TO CONVERT TO A DECIMAL

WE’LL DIVIDE. WE HAVE 28 DIVIDED BY 65.

LET’S ROUND TO 4 DECIMAL PLACES. THIS WILL BE

APPROXIMATELY 0.4308, THE 6 INDICATES WE ROUND UP. AS A PERCENTAGE,

THIS WOULD BE 43.08%. SO JUST REMEMBER THAT THIS IS

THE EXACT PROBABILITY AND THESE TWO HAVE BEEN ROUNDED

SO THEY’RE APPROXIMATIONS. I HOPE YOU FOUND THIS HELPFUL.

Thank you soooooo MUCH! Great example with visual table!!!

or got an c?