Chapter 3-probability
A certain lottery has 35 numbers. In how many different ways can 4 of the numbers be selected? (Assume that order of selection is not important.)
52369 nCr= n!/((n-r)!r!) =35!/((35-4)!4!) =1256640/4!
How many different 10-letter words (real or imaginary) can be formed from the following letters? B, B Z, Z N, N J, A, K, C
A permutation of nondistinct items without replacement is the number of ways n objects can be arranged (order matters) in which there are n1 of one kind, n2 of a second kind, and n Subscript k of a kth kind, where n=n1+n2 +..+nk. The number of such permutations is given by the following formula. 10!/2!x2!x2!=453600
game 1: 1/10 game 2: 1:10 which is better to play?
The probability of winning the first game is 1/10. The probability of winning the second game is number of wins/ number of outcomes= 1/11 Since the second probability is smaller, it would be wiser to play the first game.
Playing the game of roulette, where the wheel consists of slots numbered 00, 0, 1, 2, ..., 41 To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots.
sample space = {00, 0, 1, 2, ..., 41}. outcomes= 43
factorials (!)
used without replacement
combinations
division 5C3= (5P3)/3!
permutations
the differences 5P3= 5!/2!
A restaurant offers a $12 dinner special that has 7 choices for an appetizer, 11 choices for an entrée, and 4 choices for a dessert. How many different meals are available when you select an appetizer, an entrée, and a dessert?
7x11x4=308
0!
=1
Determine whether the events E and F are independent or dependent. Justify your answer. - E. A person living at least 70 years. F: The same person regularly handling venomous snakes - E: A randomly selected person finding cheese revolting F: Another randomly selected person finding cheese delicious - E: The unusually foggy weather in London on May 8 F: The number of car accidents in London on May 8
-E and F are dependent because regularly handling venomous snakes can affect the probability of a person living at least 70 years - E cannot affect F and vice versa because the people were randomly selected, so the events are independent. -The unusually foggy weather in London on May 8 could affect the number of car accidents in London on May 8, so E and F are dependent.
During a 52-week period, a company paid overtime wages for 16 weeks and hired temporary help for 7 weeks. During 4 weeks, the company paid overtime and hired temporary help. Complete parts (a) and (b) below. (a) Are the events "selecting a week that contained overtime wages" and "selecting a week that contained temporary help wages" mutually exclusive? (b) If an auditor randomly examined the payroll records for only one week, what is the probability that the payroll for that week contained overtime wages or temporary help wages?
a.) No b.)0.365 (30/52 +7/52 -4/52)
odds v. probability
odds of 2:3 (2/3) means probability of success is 2/5
permutation
ways in which things are ordered -fit 7 ppl into 3 chairs -how many ways can we fit 3 balls into 2 cups? 7x6x5 OR 3x2
3 things into 2 spaces
P (3,2)
P(A or B) = P(A) + P(B)
true
You randomly select one card from a standard deck. Event A is selecting a king. Determine the number of outcomes in event A. Then decide whether the event is a simple event or not.
outcomes= 4 simple event= no
T/F You toss a fair coin nine times and it lands tails up each time. The probability it will land heads up on the tenth flip is greater than 0.5.
False- You toss a fair coin nine times and it lands tails up each time. The probability it will land heads up on the tenth flip is exactly 0.5.
The probability that event A or event B will occur is P(A or B)=P(A)+P(B)−P(A or B).
False. -P(A and B)
A probability experiment consists of rolling a 20-sided die. Find the probability of the event below. rolling a prime number
0.4
A probability experiment consists of rolling a 6-sided die. Find the probability of the event below. rolling a number less than 3
2/6= 0.333
how many different groups of 3 can be selected from 5 ppl
5!/3!(5-3)!=10
If two events are mutually exclusive, they have no outcomes in common.
True
You toss a coin and randomly select a number from 0-9. What is the probability of getting tails and selecting a 9?
0.05 (1/20)
The access code for a car's security system consists of four digits. The first digit cannot be 1 and the last digit must be odd. How many different codes are available?
4,500 using 0= 9x10x10x5
Space shuttle astronauts each consume an average of 3000 calories per day. One meal normally consists of a main dish, a vegetable dish, and two different desserts. The astronauts can choose from 10 main dishes, 9 vegetable dishes, and 14 desserts. How many different meals are possible?
8190 10x9x (desserts) 2 desserts (14x13/2)
When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting?
A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!. The number of ordered arrangements of n objects taken r at a time.
T/F If two events are independent, P(A|B)equals=P(B).
True Two events A and B are independent if P(B|A)=P(B) or if P(A|B)=P(A).
A company that makes cartons finds that the probability of producing a carton with a puncture is 0.03, the probability that a carton has a smashed corner is 0.08, and the probability that a carton has a puncture and has a smashed corner is 0.002 a.) mutually exclusive? b.) If a quality inspector randomly selects a carton, find the probability that the carton has a puncture or has a smashed corner.
a.) no b.)0.108 (.03+.08-.002)
(a) List an example of two events that are independent. (b) List an example of two events that are dependent.
a.) rolling a die twice b.) Drawing one card from a standard deck, not replacing it, and then selecting another card
Identify the sample space of the probability experiment and determine the number of outcomes in the sample space. -Guessing the last digit in the price of a TV
sample space= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 outcomes= 10
You have 13 different video games. How many different ways can you arrange the games side by side on a shelf?
13!
There are 50 members on the board of directors for a certain non-profit institution. If they must elect a chairperson, first vice chairperson, second vice chairperson, and secretary, how many different slates of candidates are possible?
5527200 50x49x48x47
Outside a home, there is a 6-key keypad with letters A, B, C, D, E and F that can be used to open the garage if the correct six-letter code is entered. Each key may be used only once. How many codes are possible?
6!= 720
100!/98!
100x99 (everything else cancels out)
The odds of an event occurring are 2:6. Find (a) the probability that the event will occur and (b) the probability that the event will not occur.
a.) 2+6=8 so 2/8=0.25 b.)6/8= 0.75
A probability experiment consists of rolling a eight-sided die and spinning the spinner shown at the right (4 colors). The spinner is equally likely to land on each color. Use a tree diagram to find the probability of the given event. Then tell whether the event can be considered unusual. -Event: rolling a number less than 4 and the spinner landing on red
probability= 0.094 (3/32) unusual? No, bc it's not close enough to 0. (An event that occurs with a probability of 0.05 or less is typically considered unusual.)
Determine the number of outcomes in the event. Decide whether the event is a simple event or not. - A computer is used to select randomly a number between 1 and 9, inclusive. Event C is selecting selecting a number greater than 4.
sample space= 9 9-4=5 Event C= 5 outcomes simple event? no bc C has more than 1 outcome
Nine of the 50 digital video recorders (DVRs) in an inventory are known to be defective. What is the probability you randomly select an item that is not defective?
0.82 (50-9=41) (41/50)
Assuming that no questions are left unanswered, in how many ways can a ten-question true/false quiz be answered?
1,024 2x2x2x2x2x2x2x2x2x2=1024
Write a statement that represents the complement of the given probability. The probability of randomly choosing a tea drinker who has a college degree (Assume that you are choosing from the population of all tea drinkers.)
The probability of choosing a tea drinker who does not have a college degree
A study found that 34% of the assisted reproductive technology (ART) cycles resulted in pregnancies. Twenty-five percent of the ART pregnancies resulted in multiple births.
probability that a randomly selected ART cycle resulted in a pregnancy and produced a multiple birth= (.34x0.25)= 0.085 The probability that a randomly selected ART cycle that resulted in a pregnancy did not produce a multiple birth= 0.750 unusual? No, this is not unusual because the probability is not less than or equal to 0.05
Classify the statement as an example of classical probability, empirical probability, or subjective probability. Explain your reasoning. The probability of choosing five numbers from 1 to 36 to match five numbers drawn by the lottery is 1/376,992 almost equals 0.0000027 .
Classical because each outcome in the sample space is equally likely
Explain how the complement can be used to find the probability of getting at least one item of a particular type.
Getting "none of the items" is the set of all outcomes in the sample space that are not included in "at least one item." Using the definition of the complement of an event and the fact that the sum of the probabilities of all outcomes is 1, the following formula is obtained. P(at least one item)equals= 1−P(none of the items)
Decide if the events are mutually exclusive. Event A: Electing a president of the United StatesElecting a president of the United States Event B: Electing a female candidate
No, cuz someone who is elected to be President can be female.
Decide if the situation involves permutations, combinations, or neither. Explain your reasoning. The number of ways 19 people can line up in a row for concert tickets. Does the situation involve permutations, combinations, or neither?
Permutations. The order of the 19 people in line matters.
In the general population, one woman in ten will develop breast cancer. Research has shown that 1 woman in 650 carries a mutation of the BRCA gene. Seven out of 10 women with this mutation develop breast cancer.
The probability that a randomly selected woman will develop breast cancer given that she has a mutation of the BRCA gene= 0.7 The probability that a randomly selected woman will carry the gene mutation and develop breast cancer= (0.7x(1/650)= 0.0011 dependent
Determine which numbers could not be used to represent the probability of an event.
can't be less than 0 or greater than 1 -can be % -can be fraction -can be any decimal places (not just two)
If two events are mutually exclusive, why is P(A and B)=0?
cannot occur at the same time Two events are said to be mutually exclusive if they cannot occur simultaneously.
The number of ways a five- member committee can be chosen from 10 people.
combo- order doesnt matter- all equal positions
Researchers found that people with depression are five times more likely to have a breathing-related sleep disorder than people who are not depressed. Identify the two events described in the study. Do the results indicate that the events are independent or dependent?
2 events= depressions and breathing-related sleep disorder dependent
A combination is an ordered arrangement of objects.
The statement is false. A true statement would be "A permutation is an ordered arrangement of objects." A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!. On the other hand, a combination is a selection of r objects from a group of n objects without regard to order and is denoted by nCr.
A standard deck of cards contains 52 cards. One card is selected from the deck. (a) Compute the probability of randomly selecting a heart or diamond (b) Compute the probability of randomly selecting a heart or diamond or spade. (c) Compute the probability of randomly selecting an eight or spade
a.) 0.5 (26/52) b.) 0.75 c.) 0.308 (4+13-1)
