Download Lesson 5- Probability

STATISTICS AND THE TI - 84
Lesson 5 : Probability
PROGRAM 1: PROB1. Program for computing probabilities (no conditional probabilities given)
 PRGM  NEW
ENTER
PROGRAM
 Name: PROB1
ENTER
PROGRAM: PROB1
 : PRGM I/O 2 (Prompt)
 : Prompt A, B ENTER
 : PRGM I/O 3
 : Disp “ARE A AND B INDEPENDENT?” ENTER
 :PRGM I/O 2 (Prompt)
 :Prompt N ENTER
 :PRGM 1
 :If N 2ndTEST 1 = 1 (1 indicates: yes, 0 indicates: no) ENTER
 :PRGM 0
 :Goto C ENTER
 :PRGM I/O 3
 :Disp “ARE A AND B DISJOINT?” ENTER
 :PRGM I/O 2 (Prompt)
 :Prompt N ENTER
 :PRGM 1
 :If N 2ndTEST 1 = 1 (1 indicates: yes, 0 indicates: no) ENTER
 :PRGM 0
 :Goto D ENTER
 :PRGM I/O 2 (Prompt)
 : Prompt I ENTER (I is the probability of the intersection)
 :PRGM 0
 :Goto F ENTER
 :PRGM 9 Lbl C ENTER
 :AB STO I ENTER
 :PRGM 0 Goto F ENTER
 :PRGM 9 Lbl D ENTER
 :0 STO I ENTER
 :PRGM 0 Goto F ENTER
 :PRGM 9 Lbs F ENTER
 :A+B-I STO U: I  B STO M: I  A STO R ENTER
 :PRGM I/O 3
 :Disp “P(A AND B) IS”, I, “P(A OR B) IS”, U ENTER
 :PRGM 8 PAUSE ENTER
 :PRGM I/O 3
 : “P(A GIVEN B) IS ”, M, “P(B GIVEN A) IS”, R ENTER
 : PRGM 8 PAUSE ENTER
 :PRGM I/O 3
 : “P(ODDS IN FAVOR OF A ARE ”, A  (1-A) MATH 1 ENTER , “P(ODDS AGAISNT A
ARE ”, (1-A)  A MATH 1 ENTER 2nd QUIT
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Exercise 1. A and B are disjoint event with P(A)=.4 and P(B)=.3.
Find P(A and B), P(A or B), P(A given B) and P(B given A). What are the odd in favor and
against A?
 PRGM select PROB1
 prgmPROB ENTER
 A=? .4 ENTER
 B=? .3 ENTER
 Are A and B independent? 0 ENTER
 Are A and B disjoint? 1 ENTER
answers: P(A and B) =0, P(A or B) =.7,
 ENTER
answers: P(A given B)=0, P(B given A))=0
 ENTER answers: odds in favor of A are 2/3, odds against A are 3/2
Exercise 2. A and B are independent events with P(A)=.4 and P(B)=.3.
Find P(A and B), P(A or B), P(A given B) and P(B given A). What are the odd in favor and
against A?
 2nd ENTRY
 prgmPROB1 ENTER
 A? .4 ENTER
 B? .3 ENTER
 Are A and B independent? 1 ENTER
answer: P(A and B)=.12, P(A or B)=.58
 ENTER answers: P(A given B)=.4, P(B given A)=.3
 ENTER answers: odds in favor of A are 2/3, odds against A are 3/2
Exercise 3. If A and B are events of a sample space with P(A)=.4 and P(B)=.3 and P(A and B)=.18,
find P(A or B), P(A given B) and P(B given A). What are the odd in favor and against A?
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2nd ENTRY
prgmPROB1 ENTER
A? .4 ENTER
B? .3 ENTER
Are A and B independent? 0 ENTER
Are A and B disjoint? 0 ENTER
I? .18 ENTER
answers: P(A and B)=.18, P(A or B)=.52
 ENTER answers: P(A given B)=.6, P(B given A)=.45
 ENTER answers: odds in favor of A are 2/3, odds against A are 3/2
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Exercise 4. If P(A)=.6, P(B)=.8 and P(A and B)=.45. Find P(not A), P(A or B), P(A|B) and P(B|A). What
are the odd in favor and against A?
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PRGM
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A=?
.6
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B=?
.8 ENTER
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Are A and B independent?
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Are A and B disjoint? 0
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I= ?
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ENTER
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ENTER answers: odds in favor of A are 3/2, odds against A are 2/3
PROB1
0.45
ENTER
ENTER
ENTER
0
ENTER (0 means: no)
ENTER
answers P(A and B) is 0.4
answers P(A given B) is 0.5625
P(A or B) is 0.95
P(B given A) is 0.75
PROGRAM 2: PROB2. Program for computing with conditional
probabilities
 PRGM NEW ENTER
PROGRAM
 NAME: PROB2
ENTER
PROGRAM: PROB2
 PRGM I/O 2 (Prompt)
 Prompt A, C, D ENTER
Note: P(B given A)=C and P(B given notA)=D
 1 - A STO E: AC STO I: E D STO J ENTER
 I+J STO B: A+B-I STO U: IB STO F ENTER
 PRGM I/O 3
 Disp “P(B)=“, B, “P(A and B)=“,I ENTER
 PRGM 8 PAUSE
ENTER
 PRGM I/O 3
 Disp “P(A or B)=“, U, “P(A given B)=“, F ENTER
 PRGM 8 PAUSE
 PRGM I/O 3
 Disp “P(NEITHER A NOR B) is”, J ENTER
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: PRGM 8 PAUSE ENTER
:PRGM I/O 3
: “P(ODDS IN FAVOR OF A ARE ”, A  (1-A) MATH 1 ENTER , “P(ODDS AGAISNT A
ARE ”, (1-A)  A MATH 1 ENTER
2nd QUIT
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Exercise 5. Let A and B be events of a sample space, P(A)=.6, P(B|A)=.3
and P(B|notA)=.7. Find P(B), P(A and B), P(A or B), P(neither A nor B) and P(A|B)
 PRGM select PROB2 ENTER
 PrgmPROB2 ENTER
 A? .6 ENTER
 C? .3 ENTER
 D? .7 ENTER
answers: P(B)=.46 P(A and B)=.18
 ENTER P(P(A or B)=.88 P(A given B)=.3913043478
 ENTER P(neither A nor B) =.28
 ENTER answers: odds in favor of A are 3/2, odds against A are 2/3
Exercise 6.
Let A and B be events of a sample space, P(A)=.8, P(B|A)=.3
and P(B|notA)=.9. Find P(B), P(A or B), P(neither A nor B) and P(A|B)
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PRGM select PROB2 ENTER
PrgmPROB2 ENTER
A? .8 ENTER
C? .3 ENTER
D? .9 ENTER
answers: P(B)=.42 P(A and B)=.24
 ENTER P(P(A or B)=.98, P(A given B) = .5714285714
 ENTER P(neither A nor B) =.18
 ENTER answers: odds in favor of A are 4/1, odds against A are 1/4
Exercise 7. Find the expected value (mathematical expectation) of the values in the list {1, 4, 9,
13} with respective probabilities of {0.15, 0.18, 0.32, 0.35}
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{1, 4, 9, 13} STO 2nd L1 ENTER
{0.15, 0.18, 0.32, 0.35} STO 2nd L2 ENTER
2ndLIST MATH 5
sum(2nd L12nd L2) ENTER
answer: the expected value is 8.3
Exercise 8. Use 2nd LIST to find the expected value (mathematical expectation) of the values in
the list {1, 4, 9, 13} with respective probabilities of {0.15, 0.18, 0.32, 0.35}
 {1, 4, 9, 13} STO 2nd L1 ENTER
 {0.15, 0.18, 0.32, 0.35} STO 2nd L2 ENTER
 2ndLIST MATH 3
 mean(2nd L1, 2nd L2) ENTER
answer: the expected value is 8.3
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Exercise 9. Use STATS to find the expected value (mathematical expectation) and the standard
deviation of the values in the list {1, 4, 9, 13} with respective probabilities of {0.15, 0.18, 0.32,
0.35}
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{1, 4, 9, 13} STO 2nd L1 ENTER
{0.15, 0.18, 0.32, 0.35} STO 2nd L2 ENTER
STAT CALC 1
1-Var Stats 2nd L1, 2nd L2 ENTER
answer: the mean or expected value is 8.3 and the
standard deviation x  4.3829214
Exercise 10. Use STATS to find the expected value (mathematical expectation) and the standard
deviation of the following probability distribution
x
-7
-6
-4
3
8
12
15
19
21
25
30
36
P(x) 0.08 0.12 0.04 0.10 0.05 0.07 0.20 0.13 0.08 0.01 0.09 0.03
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{-7, -6, -4, 3, 8, 12, 15, 19, 21, 25, 30, 36} STO x ENTER
{0.08, 0.12, 0.04, 0.10, 0.05, 0.07, 0.20, 0.13, 0.08, 0.01, 0.09, 0.03} STO P ENTER
STAT CALC 1
1-Var Stats 2nd LIST X, 2nd LIST P ENTER
answer: the mean or expected value is
11.28 and the standard deviation x  12.21317322
Exercise 11. According to a mortality table the probability that Mr. Cabrera who is a 25 year old
man will die within a year is 0.008. An insurance company is selling him $100,000 policy for a
yearly premium of $1000. What is the company expected profit?
 {1000, -99000} STO  X ENTER
 {0.992, 0.008} STO  ALPHA P ENTER
 2nd LIST  MATH 3
 mean(2nd LIST X, 2nd LIST P) ENTER answer: $200
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