Binomial Distribution

Show your work for each problem.

1.
Which of the following is a condition for a binomial distribution?
1. a)Each trial must be independent
2. b)The number of trials is unlimited
3. c)Each trial must have at least 3 outcomes
4. d)The probability changes after each trial
Answer: Each trial must be independent
For a binomial distribution, each trial must be independent, meaning the outcome of one trial doesn't affect the others. Also, there must be exactly two outcomes (not 3+), a fixed number of trials, and constant probability.
2.
Calculate . Enter your answer as a number.
Answer: 15
3.
In the binomial formula , what does represent?
1. a)The number of failures
2. b)The variance
3. c)The probability of failure on each trial
4. d)The expected value
Answer: The probability of failure on each trial
In a binomial experiment, there are two outcomes: success (probability ) and failure (probability ). Since probabilities must sum to 1, the failure probability is .
4.
You flip a fair coin 4 times. What is the probability of getting exactly 2 heads?
Answer: 0.375
1. What is (number of trials)? 4
2. What is (number of successes we want)? 2
3. What is (probability of heads)? 0.5
4. Calculate 6
5. Calculate the final probability (round to 3 decimal places) 0.375
5.
If and , what is the mean of the binomial distribution?
Answer: 3
The mean is . This means on average, we expect 3 successes in 10 trials.
6.
A basketball player has a 60% free throw success rate. If she takes 5 shots, what expression gives the probability of making exactly 3?
1. a)
2. b)
3. c)
4. d)
Answer:
With , , , we get . The exponent on equals (successes), and the exponent on equals (failures).
7.
A factory produces items with a 5% defect rate. What is the probability that exactly 1 out of 10 items is defective?
Answer: 0.315
1. What is ? 10
2. What is ? 1
3. What is (defect probability)? 0.05
4. What is ? 10
5. Calculate the probability (round to 3 decimals) 0.315
8.
Calculate the standard deviation for a binomial distribution with and . Round to 2 decimal places.
Answer: 2.24
9.
Which of the following is NOT a binomial experiment?
1. a)Rolling a die and checking if it shows 6
2. b)Testing if 20 light bulbs work or fail
3. c)Drawing cards without replacement
4. d)Flipping a coin 10 times
Answer: Drawing cards without replacement
Drawing cards without replacement violates the independence condition. After drawing a card, the probability changes for the next draw because the deck has fewer cards.
10.
A vaccine is 80% effective. If 6 people are vaccinated, what is the probability that exactly 5 develop immunity?
Answer: 0.393
1. Identify , , and n=6, k=5, p=0.8
2. Calculate 6
3. Calculate 0.32768
4. Calculate 0.2
5. Final answer (round to 3 decimals) 0.393
11.
A test has 8 multiple choice questions with 4 options each. If a student guesses randomly, what is the probability of getting exactly 2 correct? Round to 3 decimal places.
Answer: 0.311
12.
A drug has a 70% success rate. If 8 patients are treated, what is the probability that at least 7 respond positively?
Answer: 0.255
1. What does 'at least 7' mean in terms of k values? k=7 or k=8
2. Calculate = (round to 3 decimals) 0.198
3. Calculate = (round to 3 decimals) 0.058
4. Add 0.255
13.
For which value of does a binomial distribution have the largest variance when ?
1. a)
2. b)
3. c)
4. d)
Answer:
The variance is . The function reaches its maximum at (this can be proven using calculus or by noting that when , which is larger than for any other value).
14.
If for a binomial distribution with , what is ? Round to 1 decimal place.
Answer: 0.4
. Taking the 5th root: . So
15.
A company knows 20% of its emails are spam. If 10 emails arrive, what is the probability of receiving at most 1 spam email?
Answer: 0.376
1. 'At most 1' means which k values? k=0 or k=1
2. Calculate 0.107
3. Calculate 0.268
4. Add 0.376
16.
A fair die is rolled 12 times. What is the expected number of sixes?
1. a)6
2. b)2
3. c)1
4. d)12
Answer: 2
The probability of rolling a six is . Expected value =
17.
In a survey, 40% of people prefer tea over coffee. If 7 people are randomly selected, what is the probability that exactly 4 prefer tea?
Answer: 0.194
1. Identify , , n=7, k=4, p=0.4
2. Calculate 35
3. Calculate 0.0256
4. Calculate 0.216
5. Final answer (round to 3 decimals) 0.194