Probability Rules
Klaus is trying to choose where to go on vacation. His two choices are: A = New Zealand and B = Alaska
0.95
The probability that he chooses either New Zealand or Alaska is P(A OR B) = P(A) + P(B) = 0.6 + 0.35 = 0.95. Note that the probability that he does not choose to go anywhere on vacation must be 0.05.
Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.
0.0789
A distribution is given as X ~ U (0, 20).
What is P(2 < x < 18)?
0.8
What is the z-score of x, when x = 1, the mean is 12 and the standard deviation is 3
-3.67
(1-12) / 3
-11/3
-3.67
What is the complement rule?
P (A') = 1 - P (A')
Helen plays basketball. For free throws, she makes the shot 75% of the time. Helen must now attempt two free throws. C = the event that Helen makes the first shot. P(C) = 0.75. D = the event Helen makes the second shot. P(D) = 0.75. The probability that Helen makes the second free throw given that she made the first is 0.85. What is the probability that Helen makes both free throws?
0.6375
P (C and D) = P (C) * P (D/C)
P (C and D) = 0.75 * 0.85 = 0.6375
It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. If 20 adult workers are randomly selected, find the probability that at most 12 of them have a high school diploma but do not pursue any further education.
0.9738
The total duration of baseball games in the major league in a typical season is uniformly distributed between 447 hours and 521 hours inclusive.
Find the mean and the standard deviation.
Mean = 484 hours
SD = 21.36 hours
The mean height of 15 to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let X = the height of a 15 to 18-year-old male from Chile in 2009 to 2010.
Find the probability that a 15-18 year old male from Chile was between 157.44 cm and 182.56 cm tall.
0.95
These values are two standard deviations away from the mean -- according to empirical rule, that corresponds to 95% of the data.
What are the three characteristics that make a distribution binomial?
1. There can only be two outcomes
2. Each trial is independent
3. There are a fixed number of trials
A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chosen randomly.
What is the probability that a member is an advanced swimmer and an intermediate swimmer?
0
You cannot be classified as an advanced swimmer and an intermediate swimmer at the same time
About 32% of students participate in a community volunteer program outside of school. If 30 students are selected at random, find the probability that between 14 and 17 of them participate in a community volunteer program outside of school.
0.0649
Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Let X = the time, in minutes, it takes a student to finish a quiz. Then X ~ U (6, 15).
Find the probability that a randomly selected student needs at least eight minutes to complete the quiz.
0.7778
In 2012, 1,664,479 students took the SAT exam. The distribution of scores in the verbal section of the SAT had a mean µ = 496 and a standard deviation σ = 114. Let X = a SAT exam verbal section score in 2012.
Find the probability that a chosen student scored between 496 and 610 on the verbal section of the SAT.
0.3413
Empirical rule!
What is the difference between mutually exclusive and non-mututally exclusive events?
Mutually Exclusive events are two events that CANNOT occur at the same time
Non-Mutually Exclusive events are two events that CAN occur at the same time
A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college play sports. Thirty of the seniors going directly to work play sports. Five of the seniors taking a gap year play sports. What is the probability that a senior is taking a gap year?
0.10
Seniors taking a gap year : 200 - 140 - 40 = 20
20/200 = 1/10 = 0.10
During a certain NBA season, a player for the Los Angeles Clippers had the highest field goal completion rate in the league. This player scored with 61.3% of his shots. Suppose you choose a random sample of 80 shots made by this player during the season. Let X = the number of shots that scored points.
Find the mean and the standard deviation of X
Mean = 49.04
SD = 4.36
The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes.
Find the 50th percentile.
16 minutes
In a uniform distribution, the mean is the same as the median. Therefore, you can use the mean formula to find this!
(11 + 21) / 2 = 32/2 = 16
Suppose X has a normal distribution with mean 25 and standard deviation five. Between what values of x do 68% of the values lie?
20 and 30!
What is the difference between independent events and dependent events?
Independent events are two events in which the probability of the first event DOES NOT IMPACT the probability of the second event.
Dependent events are two events in which the probability of the first event DOES IMPACT the probability of the second event.
Studies show that about one woman in seven (approximately 14.3%) who live to be 90 will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 2% of the time. Also suppose that in the general population of women, the test for breast cancer is negative about 85% of the time. Let B = woman develops breast cancer and let N = tests negative. Suppose one woman is selected at random.
Given that the woman has breast cancer, what is the probability that she tests negative?
0.02!
The question states it: "Suppose that of those women who develop breast cancer, a test is negative 2% of the time."
The state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. In a sample of 100 schools, find the probability of 54 of them offering fruit in the school lunches.
0.0388
Ace Heating and Air Conditioning Service finds that the amount of time a repair tech needs to fix a furnace is uniformly distributed between 1.5 and four hours. Let x = the time needed to fix a furnace. Then x ~ U (1.5, 4).
Find the 30th percentile of furnace repair times.
2.25 hours
Percentile Equation:
a + p * (b - a)
1.5 + 0.3 * (4-1.5)
1.5 + 0.3 * (2.5)
1.5 + 0.75
2.25
The scores on a college entrance exam have an approximate normal distribution with mean, µ = 52 points and a standard deviation, σ = 11 points.
Find the probability that a given student scores a 46 on the college entrance exam!
0!
The normal distribution is an example of a continuous probability distribution; therefore, we cannot find EXACT probabilities using this distribution.
What does a z-score tell you?
A z-score indicates how many standard deviations a value is away from the mean.