Transcript question - mathematicalminds.net

Statistics
Probability Distributions
Assignment 5
Example Problems
Discrete vs. Continuous
• Discrete (Countable)
– Number of students in this class
– Number of points scored in a game
• Continuous (Measurable)
– Square footage of a house
– Time to complete a job
Probability Distributions
• All probabilities
– Must add to 1
– Must be positive
– Must be between 0 and 1
• Where 0 is impossible and 1 is certain
Mean
• Finding the mean of a probability distribution
 x * P ( x )  0 * 0 . 0863
x
P(x)
0
0.0863
1
0.4935
2
0.2698
3
0.1094
4
0.0319
5
0.0091
 1 * 0 . 4935  2 * 0 . 2698  3 * 0 . 1094  4 * 0 . 0319  5 * 0 . 0091  1 . 5344
Standard Deviation
• Finding the standard deviation of a probability
distribution
 x
2
 x
2

x
P(x)
0
0.0863
1
0.4935
2
0.2698
3
0.1094
4
0.0319
5
0.0091
* P ( x )  0 * 0 . 0863  1 * 0 . 4935  2 * 0 . 2698  3 * 0 . 1094  4 * 0 . 0319  5 * 0 . 0091  3 . 2952

2
2
* P ( x )    3 . 2952  1 . 5344
2
var iance 
2
2
 0 . 940817
0 . 940817  0 . 9699  1 . 0
2
2
2
This is the variance
This is the standard deviation
Probabilities
• QUESTION: Multiple-choice questions each have five
possible answers (a, b, c, d, e), one of which is correct.
Assume that you guess the answers to three such
questions.
– Use the multiplication rule to find P(WWC), where C denotes a
correct answer and W denotes a wrong answer.
• ANSWER: Ok, notice there are 5 choices to answer
(a,b,c,d,e) so you have a 1 in 5 chance of getting the right
answer and 4 in 5 chances of getting the wrong answer.
This means
– The probability of getting the answer correct = P(C) = 1/5
– The probability of getting the answer wrong = P(W) = 4/5
– So if we get Wrong and Wrong and Correct this would be
P(WWC) = (4/5)(4/5)(1/5) = 0.128
NOTE: The word “AND” in probabilities means to multiply
Probabilities (continued)
• QUESTION: Beginning with WWC, make a complete
list of the different possible arrangements of one
correct answer and two wrong answers and then
find the probability for each entry in the list.
• ANSWER: One correct and two wrong would be
– WWC, WCW, CWW
• P(WWC) = what we got previously= 0.128
• P(WCW) = (4/5)(1/5)(4/5) = see order does not matter with
multiplication so = 0.128
• P(CWW) = (1/5)(4/5)(4/5) = see order does not matter with
multiplication so = 0.128
Probabilities (continued)
• QUESTION: Based on the preceding results,
what is the probability of getting exactly one
correct answer when three guesses are made?
• ANSWER: So this means P(WWC) OR P(WCW)
OR P(CWW)
0.128 + 0.128 + 0.128 = 0.384
NOTE: The word “OR” in probabilities means to add
Binomial Probabilities
• QUESTION: Assume that a procedure yields a
binomial distribution with a trial repeated n
times.
• Use the binomial probability of x successes
given the probability p of success on a single
trial.
• n = 9, x = 6, p = 0.65
• Find P(6)
Binomial Probabilities (calculator)
• Calculator
1) 2nd Vars
2) Scroll down to binompdf
3) The order is n, p, x so we would enter
binompdf(9, .65, 6) <--be sure to close the
parentheses
4) press enter to get 0.272 rounded
Binomial Probabilities (by hand)
• P(6) =
9!
∗ 0.656
9−6 !6!
=
∗ 0.353 =
9∗8∗7∗6∗5∗4∗3∗2∗1
∗
3 !6!
0.656 ∗ 0.353
9∗8∗7∗6∗5∗4∗3∗2∗1
∗ 0.656 ∗ 0.353
3∗2∗1∗6∗5∗4∗3∗2∗1
• now I am going to cancel 6*5*4*3*2*1 on top and bottom to be left with
9∗8∗7
∗ 0.656 ∗ 0.353
3∗2∗1
• cancel the 3 at the bottom with the 9 and cancel the 2 at the bottom with
the 8 on top to get
3∗4∗7
∗ 0.656 ∗ 0.353
1
now i put this in my calculator to get 0.272 rounded WHEW!!!
Binomial Probabilities (2nd example)
• QUESTION: A brand name has a 70%
recognition rate. If the owner of the brand
wants to verify that rate by beginning with a
small sample of 10 randomly selected
consumers, find the probability that exactly 7
of the consumers recognize the brand name.
Also find the probability that the number who
recognize the brand name is not 7.
Binomial Probabilities (2nd example)
• ANSWER: What we know
• n = 10 (number in sample)
• x = 7 (number of successes)
• p = 70% or 0.70
• We want to find the probability of exactly 7
P(7) = in your TI83 press
•
•
•
•
2nd Vars
Scroll down to binompdf
The order is n, p, x so we would enter binompdf(10,.7,7)
Press enter to get 0.267 (rounded)
• Then “not 7” would be the complement of 1 minus the
probability of 7 or
P(not 7) = 1 – P(7) = 1 – 0.267 = 0.733
Other examples
• Other examples posted in same folder