Collecting data
Randomness and probability
Sample surveys
Sampling bias
Experiments and observational studies
Q1) What role does randomness play in both observational studies and experiments?
Q2) What role does randomness play in experiments alone?
Random variables - Bernoulli, binomial, and normal
Expectation, variance, and their properties
Independent categorical variables
Probability rules
Independent events
Disjoint events
Conditional probabilities
tree diagrams
A tree diagram is most helpful when a problem seems to be describing a conditional probability.
In a tree diagram, all branches extending from the same node must add to 1.
At end end of a tree diagram, all outcomes at the far right should be disjoint (i.e. it should not be possible for any of these outcomes to occur simultaneously) and their probabilities should also sum to 1.
Suppose we are considering a random event that can be modeled as a series of independent successes or failures. Let’s say the probability of a success is 0.3 and the probability of a failure is 0.7. Find the following probabilities:
The probability of observing three successes (in a row).
The probability of observing three successes out of 10 trials.
Suppose we have a discrete random variable X with the following probability distribution:
x | 23 | -14 | 75 | 0 |
---|---|---|---|---|
Pr(X=x) | 0.20 | 0.10 | 0.15 | 0.50 |
E(X)=∑x[x⋅Pr(X=x)]
Var(X)=∑x{[x−E(X)]2⋅Pr(X=x)}
sd(X)=√Var(X)
A Diner employs three dishwashers. Al washes 40% of the dishes and breaks only 1% of those he handles. Betty and Chuck each wash 30% of the dishes and Better breaks only 1% of hers, but Chuck breaks 3% of the dishes he washes.
You’re eating at the diner one night and hear a dish break in the kitchen. What’s the probability that Chuck is on the job?
Suppose the probability that a US resident has traveled to Canada is 0.18 and the probability that a US resident has traveled to Mexico is 0.09. If probability that a US resident has traveled to both countries is 0.04, then calculate the following probabilities that a randomly selected US resident has:
traveled to Canada but not Mexico?
traveled to either Canada or Mexico?
not traveled to either country?
A company’s human resources officer reports a breakdown of employees by job type and gender as show in the table below.
Woman | Man | |
---|---|---|
Management | 6 | 7 |
Supervision | 12 | 8 |
Production | 72 | 45 |
What is the probability that a worker selected at random is:
a woman?
a woman or a production worker?
a woman if the person works in production?
a production worker if the person is a woman?
Do these data suggest that job type is independent of gender?
The probability of observing three successes (in a row) is 0.33.
The probability of observing three successes out of 10 trials is 10C3⋅0.33⋅(1−0.3)7.
E(X)=∑x[x⋅Pr(X=x)]=⋯=14.45
Var(X)=∑x{[x−E(X)]2⋅Pr(X=x)}=⋯=668.9874
sd(X)=√Var(X)=25.865