Worked solution
Decide whether an experiment has a binomial model
For each experiment, determine whether X is binomial and state its parameters where appropriate.
(a) Draw a bead ten times without replacement from a jar containing 15 red and 25 blue beads; X counts red beads obtained.
(b) Repeat with the same 15 red and 25 blue beads, but with replacement on every draw.
(c) Randomly answer 20 true/false questions; X counts correct answers.
(d) Randomly answer 20 multiple-choice questions, some with four choices and the rest with five; X counts correct answers.
CJC · Lecture · Example 1
(a) Ten bead draws without replacement
| Condition | Check |
|---|---|
| Fixed trial count | Yes: . |
| Two categories | Yes: success is a red bead; failure is a non-red bead. |
| Same success probability | No in the required independent-trial sense: the first success probability is ; after a red first draw the second is . |
| Independent trials | No: each removal changes the composition and affects subsequent draws. |
| Conclusion | X is not binomial. |
The Teacher fills the Student blanks with these values. More precisely, each draw can have the same unconditional red probability by symmetry; the conditional probabilities change with earlier results. Dependence, rather than an unequal marginal probability at every draw, is the decisive failure.
(b) Ten bead draws with replacement
| Condition | Check |
|---|---|
| Fixed trial count | Yes: . |
| Two categories | Yes: success is a red bead; failure is a non-red bead. |
| Same success probability | Yes: each time because the bead is replaced after each draw. |
| Independent trials | Yes in the source random-redraw model: a draw does not affect subsequent draws. |
| Conclusion | . |
The Teacher completions specify the same n, red/non-red classification, replacement explanation and independence. Independent uniform redraws assume that the replaced beads are mixed before the next draw; replacement alone is not a universal guarantee of independence.
(c) Twenty randomly answered true/false questions
| Condition | Check |
|---|---|
| Fixed trial count | Yes: . |
| Two categories | Yes: success is a correct answer; failure is a wrong answer. |
| Same success probability | Yes: for each question in the source random-guess model. |
| Independent trials | Yes in that model: answering one question does not affect subsequent answers. |
| Conclusion | . |
These are the Teacher completions. The calculation assumes an independent uniform guess between true and false for every question, with one correct choice. The word “randomly” alone does not specify independence for arbitrary guessing strategies.
(d) Twenty questions with four or five choices
| Condition | Check |
|---|---|
| Fixed trial count | Yes: . |
| Two categories | Yes: success is a correct answer; failure is a wrong answer. |
| Same success probability | No: it is for a four-choice question and for a five-choice question. |
| Independent trials | Yes in the source independent-guess model: each answer does not affect subsequent answers. |
| Conclusion | X is not binomial. |
The usual one-correct-choice, uniform-guess model is used. Two categories and independence are insufficient when the success probabilities differ.
Common mistakes
Equal marginal probabilities do not establish independence. Check all four binomial conditions.