Probability
There are 3 red marbles, 2 green marbles, and 1 yellow marble in a box. One marble is taken out and not replaced. A second marble is then taken out and not replaced. What is the probability both marbles that were taken out are red?
1/5, or 0.20, or 20%
A tourist in France wants to visit 12 different cities. How many different routes/ itineraries do the tourist have to choose from?
12! = 12*11*10*9*8*7*6*5*4*3*2*1 = 479001600
What are the three requirements for a probability distribution?
1. There is a numerical (not categorical) random variable x, and its number values are associated with corresponding probabilities.
2. Each probability must be between 0 and 1 (or equal to).
3. The sum of the probabilities must equal 1.
Suppose that 14% of people are left handed. If 5 people are selected at random, what is the probability that exactly 2 of them are left handed?
0.124666976
What is the critical value z0.10
1.28155
There are 10 members on a board of directors. If they must form a subcommittee of 5 members, how many different subcommittees are possible?
10C5 = 10!/(5!5!) = 252
The probability distribution is shown.
What is P(4)?
p(4) = 0.05
Because P(0) + P(1) + P(2) + P(3) + P(4) + P(5) = 1 and P(0) + P(1) + P(2) + P(3) + P(5) = 0.95,
it follows that P(4) + 0.05
If p is the probability of success of a binomial experiment, then what is the probability of failure?
q = 1 - p
A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top 40%.
212.667355
At a large factory, the employees were surveyed and classified according to their level of education and whether they smoked. The data are shown in the table.
Find the probability that a person smokes, given that they graduated college.
19/44 or 0.4318 or 43.18%
8 basketball players are to be selected to play in a special game. The players will be selected from a list of 27 players. If the players are selected randomly, what is the probability that the 8 tallest players will be selected?
1/2220075 = 4.504E-7 = 0.0000004504
Let x represent the number of times a customer visits a grocery store in a 1-week period. Assume this is the probability distribution of x:
Find the expected value of x, the average number of times a customer visits the store.
Sum of x*P(x) = mean = expected value = 1.32
A machine has 9 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working.
0.914
The meat department at a local supermarket specifically prepares its “1-pound” packages of ground beef so that there will be a variety of weights, some slightly more and some slightly less than 1 pound. Suppose that the weights of these “1- pound” packages are normally distributed with a mean of 1.00 pound and a standard deviation of .15 pound.
What percentage of the packages will weigh between .95 and 1.05 pounds?
0.26112
At a large factory, the employees were surveyed and classified according to their level of education and whether they smoked. The data are shown in the table.
If two people are selected (without replacement), find the probability that the first does not smoke and the second does smoke.
50/89 * 39/88 = 0.248978
A quiz consists of 570 true or false questions. If the student guesses on each question, what is the mean number of correct answers?
285
Six in 10 adults say lower back pain substantially limits their athletic activities
A random sample of 8 adults were asked if lower back pain was a limiting factor in their athletic activities.
a. Find the probability that all eight indicate that lower back pain was a limiting factor in their athletic activities.
b. What is the probability that at most seven individuals give lower back pain as a limiting factor in their athletic activities?
a. 0.016796
b. 1 - 0.016796 = 0.98320
The scores on a national achievement test were approximately normally distributed, with a mean of 540 and a standard deviation of 110.
a. If you achieved a score of 680, how far, in standard deviations, did your score depart from the mean?
b. What percentage of those who took the examination scored higher than you?
a. 1.272727
b. 0.101557 or 10.16%
At a large factory, the employees were surveyed and classified according to their level of education and whether they smoked. The data are shown in the table.
If one person is selected, find the probability that a person smokes and graduated college.
19/89 = 0.21348
Five cards are selected from a 52-card deck for a poker hand.
a. How many simple events are in the sample space?
b. A royal flush is a hand that contains the A, K, Q, J, and 10, all in the same suit. How many ways are there to get a royal flush?
c. What is the probability of being dealt a royal flush?
a. 52C5 = 52!/(47! * 5!) = 2598960
b. 4
c. 4/2598960 = 1/649740 = 1.539E^-6 = 0.000001539
Coffee Breaks:
a. What is the probability that a randomly selected coffee drinker would take no coffee breaks during the day?
b. What is the probability that a randomly selected coffee drinker would take more than two coffee breaks during the day?
a. 0.28 or 28%
b. P(more than 2) = P(3) = P(4) + P(5)
= 0.12 + 0.05 + 0.01 = 0.18
Human heights are one of many biological random variables that can be modeled by the normal distribution. Assume that the heights of American men have a mean of 69.5 inches and a standard deviation of 3.5 inches.
What is the probability of a randomly selected American man being over 6'0 tall? (change to inches)
Of the 43 presidents elected from 1789 through 2008, 18 were 6'0" or taller. What is the probability of getting exactly 18 out of 43 presidents over 6'0?
The probability of finding a person over 6'0 or 72" tall is 0.237525.
Getting exactly 18 out of 43 American men being taller than 6'0 has a probability of 0.004003798
The weights of certain machine components are normally distributed with a mean of 8.64 g and a standard deviation of 0.06 g. Find the two weights that separate the top 3% and the bottom 3%. These weights could serve as limits used to identify which components should be rejected. Round to the nearest hundredth of a gram.
8.53 g and 8.75 g