Math 151 , Spring 2004, Monday Day 25, April 5 Hit reload ..After class.

Chance  behavior (a random phenomenon): Unpredictable in the short run,  predictable regular pattern in the long run.

"Probability" of particular something happening: proportion of times it would happen in a very long series of independent repetitions of the phenomenon.
    (independence:  outcome of one trial (repetition) must not influence the outcome of any other.)
http://www.whfreeman.com/scc  What is probability?  1 toss at a time--settles down slowly.

Sec. 4.2 Probability Models, see Day 23
Recap: Random phenomenon,
    Sample space S:  set of all possible outcomes (no overlap of descriptions)
    Event:  any outcome or set of outcomes
    Probability model: S, and a way of assigning a probability to each event.

Probability rules:  pp. 222-3, in words, then in notation.
A an event in sample space S, P(A) is "the probability that  A occurs"
    These rules are all true for proportions in long run (Probabilities), prop.of counts, proportions of areas.
    1.  0 < P(A) < 1
    2. P(S) = 1
    3. For any event A, P(A does not occur) = 1 - P(A)
    4.  A and B are  disjoint if they have no outcomes in common (can't happen simultaneously.)
        If A and B are disjoint, their probabilities add:  P(A or B) = P(A) + P(B)

Finite sample spaces:
Assign a probability to each outcome (>0) so they add to 1.   (Sometimes equal values make sense.)
    Prob. of an event is sum of prob's of its outcomes.

Phenomenon: Flip coin twice.
    S₁ = {HH, HT, TH, TT}     S₂ = {0, 1, 2} number of heads   S₃ = {Y, N} both are heads?
Sample space  | HH | HT | TH | TT |
       Prob's | .25| .25| .25| .25|  P(tail followed by head)=?
Sample space  | 2  |    1    |  0 |  P(at least 1 tail)=?   P(1 of each) = ?
       Prob's | .25|   .50   | .25|  P(at least 1 Head)= ?  P(2Heads) = ?
Sample space  | Y  |       N      |
       Prob's | .25|     .75      |

Often the sample space is naturally expressed in numbers, thus
Random Variable:  (X, Y, Z...) Variable whose value is a numerical outcome of a random phenomenon.
 Probability distribution of X tells us what values X can take and how to assign probabilities to them.
    If X has a finite number of possible values (Discrete distributions), nothing new except notation.
     P(X < 2) is "Prob. that X is less than 2."
Flip coin twice. R.V. X = number of heads:  Distribution given by table.
             x| 2  |    1    |  0 |
       P(X=x) | .25|   .50   | .25|  P(X > 1) = ?   Words:  Prob that # heads is > 1
                                     P(X = 2) = ?         Prob that # heads is 2

Looking ahead (back)
Random variables with intervals of outcomes ("continuous") Sec. 4.2 pp. 228-232
If the sample space is an interval of values (or the whole line), the way we assign probabilities to events is with a density curve (cf. Sec. 1.3, Day 7) (remember density curves were idealizations of histograms--of repeating the "experiment" many many times)
  P(a <  X < b) = the probability that X is between a and b  is the area under the density curve, between a and b.
We declare P (X = a) = 0 , so P(a <  X < b) = P(a < X < b)

Notation: Use capital letter for the random variable, the "label" of the phenomenon.  Use small letters for particular values it can have.  But this rule is often broken--Moore uses x-bar where many would use X-bar.

B.  Y = (number you get from) the sum of two spinners. ("Triangular")
a) The probability that the sum is a number less than .6  =  P(       ?        ) =.18
b) P(Y > 1.6) =  ?     P(Y < 1.6)  =  ?        P (Y < 1) =   ?             P( 1 < Y < 1.6) =  ?
c)  P(Y > x) = .08.  Find x:  ?

Our most important probability model is the NORMAL DISTRIBUTION family.  You use the same techniques as before, only we ask "probability that one..." instead of "proportion of all..."