test 2 study pt.1
Are the events "Yes" and "No" mutually exclusive?
"Yes" and "No" are mutually exclusive because if a voter votes "Yes", they cannot also vote "No."
Give an example of two events that are mutually exclusive when a person is selected at random from a large population.
. The selected person has type A blood; the selected person has type b blood
i. The selected student only studies in the library; the selected student is wearing university sweatpants.
.The events are not mutually exclusive because if one event occurs the other event can also occur.
In most cases, it is recommended that at least how many trials be done when using a simulation to estimate a probability?
100
What is the probability of the event given below? Assume all days of the week are equally likely as birthdays. The next president of the United States was born on Monday or Saturday
2/7 chance
Is being unhappy independent of disagreeing with the statement?
Being unhappy is not independent of disagreeing with the statement because P(unhappy|disagree) is not equal to P(unhappy).
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. Refer to the graph to help decide. It is one simulation based on 100 trials.
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.
Which of the following might be a reason that an empirical probability found through a simulation does not match the theoretical probability
Both A and B are correct.
OR Probabilities with Events that are not Mutually Exclusive
If A and B are not mutually exclusive events, then P(a or b)= P(A)+P(B)- P(A and B)
a streak of 15 heads has appeared. The Law of Large Numbers says which of the following must be true?
It is equally likely that the 16th toss will be a head or a tail
When events A and B are said to be independent
Knowledge that event B occurred does not change the probability of event A occurring.
what must be done to find the probability of event A AND B. If events A and B are independent,
Multiply the probability of A and the probability of B.
Is being happy independent of agreeing with the statement?
No
inclusive OR
Outcomes that are only in A, only in B, or in both
A person is selected randomly from only the women in the group. We want to find the probability that a female responded, "has been about right." Which of the following statements best describes the question?
P(has been about right|female)
. We want to find the probability that the person selected is a male who said "hasn't gone far enough." Which of the following statements best describes the question?
P(male and responded "hasn't gone far enough")
Conditional probabilities
Probabilities where the focus is on just one group of objects and a random sample is taken from that group alone
Theoretical Probability
Probability based on comparing the number of possible favorable outcomes to the number of total possible outcomes
how can randomness be achieved
Randomness is hard to achieve without help from a computer or some other randomizing device.
Are "Republican" and "Democrat" mutually exclusive events? Give a reason for your answer.
Republican" and "Democrat" are mutually exclusive because if a voter is "Republican" they cannot be "Democrat."
Are "Republican" and "Democrat" complementary events in this data set?
Republican" and "Democrat" are not complementary because 1−P(Republican)≠P(Democrat).
if the first two rolls resulted in a 6 and a 2, the average would be 4. If the next trial resulted in a 1, the new average would be (6+2+1)/3=3. Explain how the graph demonstrates the Law of Large Numbers.
The average is variable at first, but the it settles down to the theoretical average of 3.5 at large sample sizes.
Which group is more likely to have taken a vacation in the last year: college graduates or non-college graduates? Support your answer with appropriate statistics. Select the correct choice below and fill in the answer boxes within your choice.
The college graduates are more likely to have taken a vacation because 74.8% of college graduates took a vacation in the last year, where as 48% of the non-college graduates took a vacation in the last year.
Compare the empirical probabilities to the theoretical probability, and explain what they show.
The empirical probabilities approach the theoretical probability as the sample size increases. This is an outcome expected according to the Law of Large Numbers.
A poll asked people if college was worth the financial investment. They also asked the respondent's gender. The table shows a summary of the responses. Suppose one person is selected at random from this group. Name a pair of events that are not mutually exclusive.
The event selecting a person who responded no and the event selecting a person who is female are not mutually exclusive.
ii. The selected person has black hair; the selected person has red hair.
The events are mutually exclusive because if one event occurs the other event cannot occur.
ii. The selected student drinks tea; the selected person only drinks coffee
The events are mutually exclusive because if one event occurs the other event cannot occur.
i. The selected person has brown eyes; the selected person is married.
The events are not mutually exclusive because if one event occurs the other event can also occur.
Assume a person is selected randomly from the group of people represented in the table shown below. The probability that the person says Yes given that the person is a woman is 585/723, or 80.9%. The probability that the person is a woman given that the person says Yes is 585/988 or 59.2%, and the probability that the person says Yes and is a woman is 585/1269 or 46.1%Why is the last probability the smallest
The last probability is the smallest because it is finding the proportion of people who said Yes and are a woman from all of the respondents, not just a subset of them.
If you flip a fair coin repeatedly and the first four results are tails, are you more likely to get heads on the next flip, more likely to get tails again, or equally likely to get heads or tails?
The next flip is equally likely to be heads or tails because each flip is independent of the others and the coin does not "keep track" of the past results.
A jury is supposed to represent the population. We wish to perform a simulation to determine an empirical probability that a jury of 12 people has 5 or fewer women. Assume that about 50% of the population is female, so the probability that a person who is chosen for the jury is a woman is 50%. Using a random number table, we decide that each digit will represent a juror. The digits 0-5, we decide, will represent a female chosen, and 6-9 will represent a male. Why is this a bad choice for this simulation?
The probability of selecting a digit from 0 to 5 is 6/10 or 60%, so it does not represent the probability of selecting a female juror.
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.
Which of the two pairs is more likely to have between 40% and 60% boys as grandchildren, assuming that boys and girls are equally likely as children? Why?
The second pair is more likely to have between 40% and 60% boys because the Law of Large Numbers suggests that the larger the number of repetitions, the closer the empirical probability of an event is likely to be to the true theoretical probability.
(Use your understanding) about basketball to decide whether the event that the person is shorter than six feet and the event that the person plays professional basketball are independent or associated.
The two events are associated because professional basketball players are less likely to be shorter than six feet than the average person.
typically ranges from about five feet to about six feet. Based on what you know about gender differences, if we randomly select a person, are the event that height is shorter than 6 feet and that the person is a male independent or associated? Explain.
The two events are associated because men on average are taller than women and this affects the probability of being shorter than 6 feet.
Use your knowledge about the world to decide whether the event that the person has brown eyes and the event that the person is right-handed are independent or associated.
The two events are independent because having brown eyes does not depend on being right-handed.
"the first die shows a number greater than 1 on top" independent of the event "the second die shows a number greater than 1 on top?"
The two events are independent because the result of the first die does not affect the result of the second die.
independent
Variables or events that are not associated
mutually exclusive
When two events have no outcomes in common
A driving exam consists of 25 multiple-choice questions. Each of the 25 answers is either right or wrong. Suppose the probability that a student makes fewer than 7 mistakes on the exam is 0.22 and that the probability that a student makes from 7 to 12 (inclusive) mistakes is 0.54. Find the probability of each of the following outcomes. A) A student makes more than 12mistakes B) A student makes more than 12mistakes C)A student makes at most 12 mistakes D) Which two of these three events are complementary?
a) .54+.22= .76, 1-.76=.24 B) .24+.54=.78 c) .22+.54=.76 D) More than 12 mistakes At most 12 mistakes because .more than is .24 and .76 equal 1
Determine which of the following numbers could not represent the probability of an event.
any percent's that are negative or proportions that go over the denominator because they are more than the whole, this does not apply to 0 because its a whole number
Empirical Probability
based on observations obtained from probability experiments
A poll asked people if college was worth the financial investment. They also asked the respondent's gender. The table shows a summary of the responses. Name a pair of mutually exclusive events that could result when one person was selected at random from the entire group.
he event selecting a person who responded unsure and the event selecting a person who responded no are mutually exclusive.
Law of Large Numbers
the larger the number of individuals that are randomly drawn from a population, the more representative the resulting group will be of the entire population
Give an example of two events that are not mutually exclusive when a student is selected at random from a large college population.
the selected student only studies in thier dorm; he selected student rides thier bike to class
complementary probabilities
two mutually exclusive outcomes with a combined probability of 1
Are the events "Yes" and "Democrat" mutually exclusive?
"Yes" and "Democrat" are not mutually exclusive because a voter can vote "Yes" and be a "Democrat."
Are the events "Yes" and "No" complementary events for this data set?
"Yes" and "No" are complementary because 1−P(Yes)=P(No).
given that
Often, conditional probabilities are worded