Counting the wins in a fixed number of goes

Mathematics · Probability · ages 16-17

Name ______________________ Date ____________

In a binomial model, every ordered arrangement with the same number of successes has the same probability.
Circle one: True False
A fair coin is flipped independently 3 times. What is the probability of no heads at all? Give an exact decimal.
Answer: ______________
Four independent trials each have success probability 0.3. What is the probability of the exact order success, nonsuccess, success, nonsuccess? Give an exact decimal.
Answer: ______________
Which of these is NOT one of the four conditions for a binomial model?
- the number of trials is fixed before you start counting them
- each trial result is classified as success or nonsuccess
- the number of successes is known in advance of the trials
- the probability of a success is the same on every trial
A box contains 10 phones, all faulty. Eight are selected without replacement. Their faulty count has exactly the distribution B(8,1).
Circle one: True False
Six independent computer trials record success or not. Their success probabilities are 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6. Which binomial condition fails?
- The number of trials is not fixed because each trial has a different p.
- There are more than two outcomes because six probabilities are listed.
- The success probability varies across trials instead of sharing one p.
- The trials are dependent because different probabilities force dependence.
A fair coin is flipped independently 4 times. What is the probability of 4 heads? Give an exact decimal.
Answer: ______________
Which situation fits a binomial model?
- Draw 4 pupils uniformly without replacement from 30, including 12 glasses wearers; count glasses wearers.
- Roll the same fair die independently 20 times; count sixes and classify every other face as a nonsuccess.
- Measure 20 pupils' heights in centimetres; keep each numerical height instead of using two categories.
- Draw red and blue counters until red appears; count how many draws were needed to reach that first red.
A quiz has 12 questions, each with 4 options and exactly one correct option. One option is chosen uniformly and independently for each question. Is the correct-answer count binomial?
- No: uniform guesses do not count as repeated trials of an experiment.
- No: four answer options prevent a two-category success-or-not model.
- Yes: twelve independent correct-or-not trials share success probability 0.25.
- Yes: guessing makes every possible total from zero to twelve equally likely.
If a binomial model does not fit, the arithmetic will produce an error rather than an answer.
Circle one: True False

Answer key

For grown-ups. Fold this page away before handing over the rest.

Counting the wins in a fixed number of goes W1-mt_X2kVGK67Md-s1

True · Independence and constant p give the same factors for each arrangement with a fixed success count. Reordering those factors does not change their product.
0.125 · Only one arrangement gives no heads, tails tails tails, and it has probability 0.5 × 0.5 × 0.5 = 0.125.
0.0441 · Multiply the probabilities in the specified order: 0.3 × 0.7 × 0.3 × 0.7 = 0.0441. Only one order is requested.
the number of successes is known in advance of the trials · Knowing the success count beforehand is not a binomial condition. X counts successes. The conditions are fixed trials, two outcome categories, constant success probability and independence.
True · All eight selected phones must be faulty, so the count is always 8. B(8,1) gives the same constant count. This is an exception to the mixed-population sampling issue.
The success probability varies across trials instead of sharing one p. · Six trials are fixed, success or not gives two categories, and independence is stated. The success probabilities differ, so the constant-probability condition fails.
0.0625 · Only one arrangement gives four heads, and its probability is 0.5 to the fourth power = 0.0625.
Roll the same fair die independently 20 times; count sixes and classify every other face as a nonsuccess. · The die experiment has fixed trials, six or not-six outcomes, constant p = 1/6 and independence. The other situations have dependent selections, ungrouped numerical outcomes or a variable trial count.
Yes: twelve independent correct-or-not trials share success probability 0.25. · The counted categories are correct and not correct. There are 12 fixed independent trials. One correct option among four equally likely options gives p = 0.25.
False · A calculation can return a number even when exact binomial conditions fail. Check the situation separately. The result may be a useful approximation, but it is not automatically an exact probability.