Explain in detail and give an original example of the four types of probability.
1) Marginal probability
2) Union probability
3) Joint probability
4) Conditional probability
All of these involve probabilities of two (or more) variables or events. The two variables (let's call them x and y) can be independent or not.
1) Marginal probability P(x)
This is the probability of x, over all possible y's. If x is some event, you would find it by dividing the number of times x occurred by the total number of observations. You wouldn't consider what y did during that same time.
2) Union probability P(x or y)
This is the probability that either x or y or both will occur. If x and y are events A and B, then the union probability is the sum of the two events minus the joint probability:
P(A or B) = P(A) + ...
Four types of probability are examined. The marginal, union, joint and conditional probability is analyzed.
The probability of a successful firing is, on any test, 0.95.
1. Five cruise missiles have been built by an aerospace company. The probability of a successful firing is, on any test, 0.95. Assuming independent firings, what is the probability that the first failure occurs on the fifth firing?
2. A lot of 25 color television tubes is subjected to an acceptance testing procedure. The procedure consists of drawing five tubes at random, without replacement, and testing them. If two or fewer tubes fail, the remaining ones are accepted. Otherwise, the lot is rejected. Assume the lot contains four defective tubes. What is the exact probability of lot acceptance?
3. Aircrew escape systems are powered by a solid propellant. The burning rate of this propellant is an important product characteristic. Specifications require that the mean burning rate must be 50 cm/s. We know that the standard deviation of burning rate is σ = 2 cm/s. The experimenter decided to specify a type I error probability or significance level of α = 0.05. He selects a random sample of n = 25 and obtains a sample average burning rate of = 51.3 cm/s.
a. Test the hypothesis that the mean burning rate is 50 cm/s.
b. What is the P-value for this test?
c. Construct a 95% two-sided confidence interval on mean burning rate.
4. An automatic filling machine is used to fill bottles with liquid detergent. A random sample of 20 bottles results in a sample variance of fill volume of = 0.0153 square fluid ounces. If the variance of fill volume exceeds 0.01 square fluid ounces, an unacceptable proportion of bottles will be under and overfilled. Is there evidence in the sample data to suggest that the manufacturer has a problem with under- and overfilled bottles? Use α = 0.05, and assume that fill volume has a normal distribution.
5. An experiment was performed to determine the effect of four different chemicals on the strength of a fabric. These chemicals are used as part of the permanent press finishing process. Five fabric samples were selected, and a randomized complete block design was run by testing each chemical type once in random order on each fabric sample. The data are shown below.
Chemical Type Fabric Strength
1 2 3 4 5
1 1.3 1.6 0.5 1.2 1.1
2 2.2 2.4 0.4 2.0 1.8
3 1.8 1.7 0.6 1.5 1.3
4 3.9 4.4 2.0 4.1 3.4
a. Perform an analysis of variance to test whether the differences among the sample means of the chemical types are significant. Use α = 0.05
b. Construct box plots of fabric strengths by chemical types.
c. Construct a normal probability plot of the residuals from this experiment. Does the assumption of a normal distribution for fabric strength seem reasonable?