Basic Terms of Probability
Section 3.2

(Read 3.1 to get acquainted with casino games.)

 

Definitions:

 

Experiment:  a process by which an outcome is obtained, i.e., rolling a die.

 

Sample space: The set S of all possible outcomes of an experiment.

i.e. the sample space for a die roll is {1, 2, 3, 4, 5, 6}

 

Event:  Any subset E of the sample space.  i.e. Let 

        E1 = An even number is rolled.

        E2 = A number less than three is rolled.

 

Probability and Odds:

The probability of an event is a measure of the likelihood that the event will occur.

If an experiment’s outcomes are equally likely to occur, then the probability of an event E is the number of outcomes in E divided by the number of outcomes in the Sample Space.      

        P(E) = n(E)/n(S)

 

This chapter only discusses experiments with equally likely outcomes.

Note: Probability   =   (success)
                                    (total)

 

Odds in favor of an event E are the number of ways the event can occur compared to the number of ways the event can fail.

        O(E) = n(E) : n(E’)                   Odds = (success) : (failure)

 

House odds               vs.       True odds

What the casino pays           vs.       what you should get

 

Example 1:  

On a Roulette wheel, find:

 

a)     the probability of getting a red number

b)     the odds of getting a red number

c)      the probability of getting number 20

d)     the odds of getting number 20

e)     the probability of getting a number between 1 and 12 inclusive

f)        the odds of getting a number between 1 and 12 inclusive

 

a) 18/38 = 9/19          b) 18: 20 = 9: 10

 

c) 1/38                        d) 1: 37

 

e) 12/38 = 6/19          f )  12 : 26 = 6 : 13

 

 

Probability can be found theoretically or empirically.  Up to now, we have used theory.

Empirical means to scientifically do each experiment and record the observations.

If we flip a coin and record how many heads comes up, then it is called relative frequency of heads.  The results may not be exactly the ˝ probability that theory provides, but if the coin is flipped a large number of times, the relative frequency will come close to the ˝ probability.  This is called the Law of Large Numbers.

 

Gregor Mendel’s experiments in genetics first used probability.  Genes are recessive and dominant.  Using pea plants, he studied the outcome of cross-pollinating red and white flower plants.  At the time, predictions were that the flowers would turn out pink.  He made sure the flowers were pure color,  1st generations of white and red.

 

Using a Punnett Square:                                          R                   R

R = red,  w = white                                        w         ( R, w)             (R, w)

Results were all Red Flowers                      w         ( R, w)             (R, w)

For the 2nd generation.

Thus Red was dominant and white recessive.

 

                                                            R                     w

Results:  3 Red, 1 white flower.                   R         (R, R)              (R, w)

                                                                        w         (R, w)              (w, w)

                                    Third generation:

One true Red, 2 red with white carriers, 1 true white.

 

Applications of probability in genetics include predictions, likelihoods of passing genetic material to offspring.  Decisions can be made whether to have children if a person is a carrier of a fatal disease.

 

Huntington’s disease is caused by a dominant gene.  Woody Guthrie, famous folksinger-song-writer died of the disease at age 55.  His son Arlo Guthrie did not want to be tested for the disease, there is no cure. 

 

Sickle-cell anemia is another disease, less fatal, but dangerous.

 


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