Consider a short multiple choice quiz with three items, and eachitem has four choices (only one of the choices is correct). Supposethat you are taking this quiz but you are completely and utterlyunprepared for it. That means that you only option for the quiz isto guess the answers. Suppose you are thinking about the firstitem: what's the probability that you'll answer it incorrectly(hint: think about how many options there are, and how many of themare wrong)?
Now, list the \"sample space\" of all possible outcomes on thisexam. Hint: here is one possible outcome: Correct, Correct, Wrong(or, CCW for short).
Using the format in #2 above, we actually haven't listed theentire sample space, because there are multiple ways to answer eachitem incorrectly! In order to calculate the probability of aparticular outcome, we either need to list every single way toanswer a problem wrong, or we can use the multiplication rule.These outcomes are all independent because guessing on problem #1has no effect on your guess for problems #2 or #3. For example,P(Correct on the 1st, and Correct on the 2nd, and Wrong on the 3rd)= P(Correct on the 1st) * P(Correct on the 2nd) * P(Wrong on theThird). Find P(CCW).
Now, use your list from #2 and write the probability for eachoutcome next to it, using the same type of calcuation you did in#3.
Next, use your work from #4 to fill in the following table withthe relevant probabilities, where x represents the number of itemsyou answer correctly. Notice that the number of correct items willsometimes include more than one of the listed outcomes (what do youdo with those numbers?)
In #5 above, you should have a probability distribution (whichmeans that all probabilities are between 0 and 1, and the sum ofthe probabilities is 1). Check to make sure this is correct, fixany errors, and then explain why this situation fits the criteriafor a binomial distribution.
Using the table you made in #5, determine the probability ofgetting at most 2 items correct on the quiz.
We don't need to calculate each probability by hand. It can bedone in a spreadsheet: Using excel or sheets, you can type thecommand \"BINOM.DIST\" (in sheets, there is no dot) to help computethese probabilities. For example, the command\"=BINOM.DIST(0,3,.25,0)\" tells you the probability of getting 0correct (# of \"successes\") out of three (# of trials) where theprobability of getting one item correct is .25 (probability of a\"success\"), and we want exactly that number of correct items (\"0\"at the end of the command refers to the fact that we aren't addingup any probabilities, that the \"cumulative\" feature of the commandis turned off). Verify, using this command, that each of the valuesin your table above is correct.
Now, you can also check the calculation you did in #7, by typingthe following command: \"BINOM.DIST(2,3,.25,1)\", which means youwant to get two correct (# of successes) out of three items (# oftrials), where the probability of getting one item correct is .25(probability of a success), and we are adding up the probabilitiesfor all values below 2 as well (the cumulative spot is turned onwhen you type \"1\").
Finally, create a histogram of the distribution and copy it intoyour response.
