Probability
Probability – Basic Concepts
Probability
The study of the occurrence of random events or phenomena.
It does not deal with guarantees, but with the likelihood of an occurrence of an event.
Experiment:
- Any observation or measurement of a random phenomenon.
Outcomes:
- The possible results of an experiment.
Sample Space:
- The set of all possible outcomes of an experiment.
Event:
- A particular collection of possible outcomes from a sample space.
Probability – Basic Concepts
Example:
If a single fair coin is tossed, what is the probability that it will land heads up?
Sample Space:
Event of Interest:
P(heads) = P(E) =
The probability obtained is theoretical as no coin was actually flipped
Theoretical Probability:
P(E) =
number of favorable outcomes
S = {h, t}
E = {h}
1/2
total number of outcomes
n(E)
n(S)
=
Probability – Basic Concepts
Example:
A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. What is the probability that it lands on it top?
P(top) =
The probability obtained is experimental or empirical as the cup was actually flipped.
Empirical or Experimental Probability:
P(E)
number of times event E occurs
number of times the experiment was performed
number of top outcomes
total number of flips
10
100
=
1
10
=
͌
Probability – Basic Concepts
Example:
There are 2,598,960 possible five-card hand in poker. If there are 36 possible ways for a straight flush to occur, what is the probability of being dealt a straight flush?
P(straight flush) =
This probability is theoretical as no cards were dealt.
number of possible straight flushes
total number of five-card hands
36
2,598,960
=
0.0000139
=
Probability – Basic Concepts
Example:
A school has 820 male students and 835 female students. If a student is selected at random, what is the probability that the student would be a female?
P(female) =
This probability is theoretical as no experiment was performed.
number of possible female students
total number of students
835
820 + 835
=
0.505
=
835
1655
=
167
331
P(female) =
Probability – Basic Concepts
The Law of Large Numbers
As an experiment is repeated many times over, the experimental probability of the events will tend closer and closer to the theoretical probability of the events.
Flipping a coin
Spinner
Rolling a die
Probability – Basic Concepts
Odds
A comparison of the number of favorable outcomes to the number of unfavorable outcomes.
Odds are used mainly in horse racing, dog racing, lotteries and other gambling games/events.
Odds in Favor: number of favorable outcomes (A) to the number of unfavorable outcomes (B).
Example:
A to B
A : B
What are the odds in favor of rolling a 2 on a fair six-sided die?
1 : 5
What is the probability of rolling a 2 on a fair six-sided die?
1/6
Probability – Basic Concepts
Odds
Odds against: number of unfavorable outcomes (B) to the number of favorable outcomes (A).
Example:
What are the odds against rolling a 2 on a fair six-sided die?
B to A
B : A
5 : 1
What is the probability against rolling a 2 on a fair six-sided die?
5/6
Probability – Basic Concepts
Odds
Two hundred tickets were sold for a drawing to win a new television. If you purchased 10 tickets, what are the odds in favor of you winning the television?
Example:
200 – 10 =
10 : 190
What is the probability of winning the television?
10/200
190
Unfavorable outcomes
10
Favorable outcomes
=
1 : 19
1/20
=
=
0.05
Probability – Basic Concepts
Converting Probability to Odds
The probability of rain today is 0.43. What are the odds of rain today?
Example:
100 – 43 =
43 : 57
The odds for rain today:
57
Unfavorable outcomes:
P(rain) = 0.43
Of the 100 total outcomes, 43 are favorable for rain.
57 : 43
The odds against rain today:
Probability – Basic Concepts
Converting Odds to Probability
The odds of completing a college English course are 16 to 9. What is the probability that a student will complete the course?
Example:
16 : 9
The odds for completing the course:
P(completing the course) =
Favorable outcomes + unfavorable outcomes = total outcomes
16 + 9 = 25
= 0.64