UCLA Probabilities for Outcomes of A Binomial Process & Permutations Questions

Table 2 in your book gives probabilities for outcomes of a binomial process.  If, for some reason, you don’t have the book, the same table can be found here (https://www.sjsu.edu/people/saul.cohn/courses/stats/s0/BinomialProbabTable.pdf). Before you freak out, let’s just go over what this means.  A coin flip is a binomial process — the coin can land heads or tails.  If I flip that coin five times in a row, I can get 0, 1, 2, 3, 4, or 5 heads.  There are 32 different permutations that all fit into one of the 6 combinations (0 to 5 heads).  There is only one way to get zero heads and that’s by flipping tails five times in a row.  So the probability of getting 0 heads in 5 trials of a binomial test where the odds are 50-50 is 1/32, or .03125.  Take a look at the binomial table.  Take a look at the section that says 5 to the far left.  The first row is 0.  If you scan over to the column labeled .50, you will see this same probability — .031.

Now, there are five ways to get one heads.  You can flip:

HTTTT

HTHHH

HHTHH

HHHTH

HHHHT

So the probability of getting exactly one heads is 5/32 or .15625.  Again, take a look at the table and see that this is the answer the table gives as well (though rounded).  Okay, now you are ready to answer the first few questions in this homework.

Q1:

Donald Trump is currently trailing in polls and looks to be on course to lose the election.  However, he has a roughly 45% chance of winning in each of six crucial states: North Carolina, Florida, Arizona, Pennsylvania, Michigan, and Wisconsin. Given that his probability of winning in each state is .45, what is his probability of winning all six states (and thereby, a second term in office)?

Select one:

a. .008

b. .136

c. .028

d. .303

e. .186

f. .061

g. .278

Q2:

President Trump only really needs to win four of these six states to probably win the election.  What is the probability he will win at least four of these states?  I’m not asking the probability of him winning EXACTLY four of these six states; I’m asking the probability of him winning AT LEAST four of these six states.

Select one:

a. .069

b. .255

c. .186

d. .745

e. .931

Q3:

Let’s say that among SF State students, the probability of contracting covid-19 if exposed is .65.  If I am currently contagious and I meet with 8 other professors, what is the probability that all 8 will contract covid-19?

Select one:

a. .188

b. .022

c. .279

d. .081

e. .137

f. .032

g. .003

h. .259

Q4:

What’s the probability I will infect at least half (4) of the group of eight professors?

Select one:

a. .188

b. .294

c. .05

d. .02

e. .706

f. .50

Q5:

I pick five of you to play a dice game with me.  Each of us gets one number of the die: 1-6.  If the die lands on a three, for example, all five other plays have to pay person three a dollar.  After 18 rolls, I have won five times.  Should you accuse me of cheating?  And if after 180 rolls, I have won fifty times, should you accuse me of cheating?  Explain your answers.