Statistics Chapter 5
Often, conditional probabilities are worded with what phrase?
"given that"
Probabilities where the focus is on just one group of objects and a random sample is taken from that group alone are called what?
Conditional probabilities
In statistics, what is true of randomness?
Randomness is hard to achieve without help from a computer or some other randomizing device.
If an experiment with a random outcome is repeated a large number of times, the empirical probability of an event is likely to be close to the true probability. This mathematical theorem is called what?
The law of large numbers
Imagine flipping a fair coin many times. Explain what should happen to the proportion of heads as the number of coin flips increases. Which of the following is the best explanation to what should happen to the proportion of heads as the number of coin flips increases?
The proportion should get closer to 0.5 as the number of flips increases.
The sample space of a random experiment is what?
The set of all possible and equally likely outcomes of the experiment
Probabilities that are based on short-run relative frequencies are called what?
Empirical probabilities
The percentage of left-handed people in a certain country is estimated to be 9%. Women are about six times as likely to be left-handed as men. Are gender and handedness independent or associated? Explain.
Gender and handedness are associated because women are more likely to be left-handed than men.
Because they are generated by a seed value that starts the random sequence, computer-generated random numbers are sometimes called what?
Pseudo-random numbers
Experiments used to produce empirical probabilities are called what?
Simulations
Given the event "a die lands with a 6 on top", which of the following is the complement of this event?
The die lands with a 1, 2, 3, 4, or 5 on top
Assume the only grades possible in a history course are A, B, C, or lower than C. The probability that a randomly selected student will get an A in the course is 0.40, the probability that a student will get a B in the course is 0.24, and the probability that a student will get a C in the course is 0.17. What is the probability that a student will get a grade lower than a C?
The probability that a student will get a grade less than C is 0.19.
Assume the only grades possible in a history course are A, B, C, or lower than C. The probability that a randomly selected student will get an A in the course is 0.40, the probability that a student will get a B in the course is 0.24, and the probability that a student will get a C in the course is 0.17. What is the probability that a student will get an A OR a B OR a C?
The probability that a student will get an A OR a B OR a C is 0.81.
Suppose a person is chosen at random. Use your knowledge about the world to decide whether the event that the person has green eyes and the event that the person is right dash handed are independent or associated.
The two events are independent because having green eyes does not depend on being right dash handed.
When events A and B are said to be independent, what does that mean?
Knowledge that event B occurred does not change the probability of event A occurring.
If events A and B are independent, what must be done to find the probability of event A AND B?
Multiply the probability of A and the probability of B.
If 29% of Americans households own one or more dogs and 43% own one or more cats, then from this information, is it possible to find the percentage of households that own a cat OR a dog? Why or why not?
No, because the event of owning a dog and the event of owning a cat are not mutually exclusive. Therefore, to find the percentage of people that own a cat or a dog, it is necessary to know the percentage of people that own a cat and a dog.
Statistics and probability use the "inclusive OR". This means that referring to outcomes A OR B is referring to what?
Outcomes that are only in A, only in B, or in both
Probabilities are always numbers between and including what numbers?
0 and 1
Betty and Jane are gambling. They are cutting cards (picking a random place in the deck to see a card). Whichever one has the higher card wins the bet. If the cards have the same value, they try again. Betty and Jane do this 100 times. Tom and Bill are doing the same thing but only betting 10 times. Is it Bill or Betty who is more likely to end up having very close to 50% wins? Explain.
Betty is more likely to end up having close to 50% wins as she is betting more times and the Law of Large Numbers says that the more times a random experiment is repeated the closer it comes to the true probability.
Variables or events that are not associated are called what?
Independent
Suppose a weather forecaster says the probability that it will rain on Saturday is 29% and the probability that it will rain on Sunday is 36%. From this information, is it possible to find the probability that it will rain on Saturday or Sunday (or both)? Why or why not?
No, because the event of raining on Saturday and the event of raining on Sunday are not mutually exclusive. Therefore, to find the probability that it will rain on Saturday or Sunday, it is necessary to know the probability that it will rain on both Saturday and Sunday.
The table below summarizes results from a survey that asked about political party affiliation and self-described political orientation. (Dem means Democrat, and Rep means Republican.) If a person is selected at random, is the event that he or she is a Republican independent of the event that he or she is liberal?
The event that he or she is a Republican is not independent of the event that he or she is liberal, because the probabilities are not equal.
A single die is rolled. Find the probability of rolling an odd number or a number less than 3.
The probability is 2/3
Assume the only grades possible in a history course are A, B, C, or lower than C. The probability that a randomly selected student will get an A in the course is 0.40, the probability that a student will get a B in the course is 0.24, and the probability that a student will get a C in the course is 0.17. What is the probability that a student will get an A OR a B?
The probability that a student will get an A OR a B is 0.64
A friend flips a coin 10 times and says that the probability of getting a head is 60% because he got six heads. Is the friend referring to an empirical probability or a theoretical probability? Explain.
This is an example of empirical probability because it is based on an experiment.
Seat belt use in a state was estimated at 78%, which means 78% of people use their seat belts. Suppose two independent drivers have been randomly selected. a. What is the probability that both of them are using a seatbelt? b. What is the probability that neither of them is using a seatbelt? c. What is the probability that at least one is using a seatbelt?
a. The probability that both of them are using a seatbelt is 0.6084. b. The probability that neither of them is using a seatbelt is 0.0484. c. The probability that at least one of them is using a seatbelt is 0.9516. c is the complement of b c+b=1 1-0.0484=0.9516
In most cases, it is recommended that at least how many trials be done when using a simulation to estimate a probability?
100
If the probability that it will rain tomorrow is 0.30, the probability that it will not rain tomorrow is what?
0.70
Which of the following is the probability that something in the sample space will occur?
1
When two events have no outcomes in common, they are called what?
Mutually exclusive
The table below summarizes results from a survey that asked about political party affiliation and self-described political orientation. (Dem means Democrat, and Rep means Republican.) A person is selected randomly from the sample summarized in the table. We want to find the probability that a liberal person is a Democrat.
P(Democrat | Liberal)
What do we call the law that causes this settling down of the proportion?
The law of large numbers
A person was trying to figure out the probability of getting two heads when flipping two coins. He flipped two coins 10 times, and in 5 of these 10 times, both coins landed heads. On the basis of this outcome, he claims that the probability of two heads is 5/10, or 50%. Is this an example of an empirical probability or a theoretical probability? Explain.
This is an example of empirical probability because it is based on an experiment.
A Monopoly player claims that the probability of getting a 4 when rolling a six-sided die is one sixth because the die is equally likely to land on any of the six sides. Is this an example of a theoretical probability or an empirical probability? Explain.
This is an example of theoretical probability because it is not based on an experiment.
Consider the following categories of people, assuming that we are talking about all the people in a certain country. Category 1: People who own a cat Category 2: People who own a dog Category 3: People who own a cat OR own a dog Category 4: People who own a cat AND own a dog a. Which of the four categories has the most people? b. Which category has the fewest people?
Which of the four categories has the most people? Category 3: People who own a cat OR own a dog Which category has the fewest people? Category 4: People who own a cat AND own a dog
