Section 2.3, 2.4, 3.1
MATH 166:503
February 24, 2015
Topics from last class: distinguishable permutations, combinations, number of combinations
of r objects from a set of n objects.
2
COUNTING AND PROBABILITY
2.3
Probability applications of counting principles
ex.(again) A fair coin is tossed 6 times. Find the probability of obtaining exactly 3 heads.
What is the probability that the first three are heads?
Notes:
1
ex. If A, B, C, D, and E go to the movies and randomly select 5 adjacent seats, what is the
probability that A sits next to B?
ex.(again) A bowl contains 6 blueberries and 9 cranberries. If we select two without replacement,
what is the probability that both are cranberries?
What is the probability with replacement?
ex. 23 of 30 oranges collected from a tree are ripe. 10 are randomly selected to be given to
neighbors. What is the probability that at least 8 of those we gave to neighbors are ripe?
Notes:
2
2.4
Bernoulli Trials
ex. If you are taking a 5 question multiple choice test with 4 choice options for each question. If
you pick answers at random, what is the probability of answering 3 questions correctly?
ex. If we perform 9 repeated Bernoulli trials with p = 0.23, what is the probability of getting 5
successes?
Notes:
3
ex. It is estimated that only 56% of Americans can perform basic swimming skills (http://time.com/106912/redcross-swimming-campaign/). A sample of 25 Americans are randomly selected. What is the probability that 10 are able to perform these basic skills?
What is the probability that at least 3 can swim?
What is the probabilty that at least 3 but fewer than 6 can swim?
ex. If we flip a coin 9 times where the probability of flipping a heads is 0.45, what is the probability
that the first 4 flips are heads and the last flip is a tails?
Notes:
4
3
PROBABILITY DISTRIBUTIONS AND STATISTICS
3.1
Random Variables and Histograms
ex. If we are sampling cartons of half a dozen eggs and our experiment is to look which are broken.
A random variable for this experiment is the number which are broken. What are the possible
values of this random variable?
ex. What type of random variable are the following random variables?
a.
time spent studying math each week
b.
hours spent studying math each week
c.
the number of math problems answered before getting one wrong
ex. If we roll a 3-sided die twice and look at the top faces. A random variable for this experiment
is the sum of the two die. What are the possible values of this random variable? What are the
outcomes that correspond to each value?
Find the probability distribution for this random variable.
Notes:
5
ex. Roll a 12-sided die. Let X be the random variable given by 1 if the die is a multiple of 4 and 0
otherwise. Find the probability distribution for X.
ex. 2 cards are drawn from a standard desk of cards. Let X be the number of cards drawn that
are hearts. Draw the histogram for X.
Notes:
6
ex. Find the following probabilities corresponding to the histogram.
a.
P (X = 9)
b.
P (4 ≤ X < 7)
c.
P (X = 3 or X = 8)
Notes:
7