Binomial Probability

Show your work for each problem.

1.
Which of the following is a requirement for a binomial experiment?
1. a)Outcomes can vary each trial
2. b)At least 100 trials required
3. c)Probability changes between trials
4. d)Trials must be independent
Answer: Trials must be independent
Independence is crucial: the outcome of one trial must not affect another. The probability must stay constant, and there must be exactly two outcomes per trial.
2.
What is ?
Answer: 6
3.
In the formula , what does represent?
1. a)Number of trials
2. b)Number of successes
3. c)Probability of success on each trial
4. d)Probability of failure
Answer: Probability of success on each trial
is the probability of success on each individual trial. It must remain constant throughout all trials.
4.
If , what is ?
Answer: 0.7
. This represents the probability of failure on each trial.
5.
A fair coin is flipped 4 times. What is the probability of getting exactly 2 heads?
Answer: 0.375
1. What is (total trials)? 4
2. What is (successes wanted)? 2
3. What is ? 6
4. What is ? 0.25
5. What is for failures? 0.25
6. Multiply all parts: 0.375
6.
Calculate .
Answer: 20
7.
Which scenario is NOT a binomial experiment?
1. a)Rolling a die 6 times and counting sixes
2. b)Testing if 8 light bulbs work
3. c)Flipping a coin 10 times
4. d)Drawing 5 cards from a deck without replacement
Answer: Drawing 5 cards from a deck without replacement
Drawing cards without replacement changes the probability after each draw, violating the constant probability requirement.
8.
A basketball player makes 60% of free throws. What is the probability of making exactly 3 out of 5 free throws?
Answer: 0.3456
1. What is (success probability)? 0.6
2. What is (failure probability)? 0.4
3. What is ? 10
4. What is ? 0.216
5. What is ? 0.16
6. Calculate 0.3456
9.
What is ? Round to 4 decimal places.
Answer: 0.2401
10.
In a binomial experiment with and , which value of has the highest probability?
1. a)
2. b)
3. c)
4. d)
Answer:
For , the binomial distribution is symmetric, peaking at . This is where is largest.
11.
A multiple choice test has 8 questions with 4 options each. If you guess randomly, what is the probability of getting exactly 2 correct?
Answer: 0.3115
1. What is the probability of guessing one question correctly? 0.25
2. What is ? 28
3. What is ? 0.0625
4. What is ? Round to 4 decimals. 0.1780
5. Calculate 0.3115
12.
Calculate .
Answer: 210
13.
A drug has an 80% success rate. If 6 patients take it, what is the probability that ALL 6 respond positively?
1. a)
2. b)
3. c)
4. d)
Answer:
For all successes (),
14.
A quality inspector finds defects in 5% of products. In a sample of 10 products, what is the probability of finding exactly 1 defect?
Answer: 0.3151
1. What is (defect probability)? 0.05
2. What is ? 10
3. What is ? 0.05
4. What is ? Round to 4 decimals. 0.6302
5. Calculate 0.3151
15.
What is the probability of getting NO successes in 4 trials if ? Round to 4 decimal places.
Answer: 0.2401
16.
What is ?
1. a)1
2. b)0
3. c)Undefined
4. d)5
Answer: 0
for any . There is exactly one way to choose nothing: choose nothing!
17.
What is ?
Answer: 1
for any . There is exactly one way to choose all items: take them all!
18.
A vaccine is 90% effective. In a group of 7 vaccinated people, what is the probability that exactly 6 are protected?
Answer: 0.372
1. What is ? 7
2. What is ? Round to 4 decimals. 0.5314
3. What is ? 0.1
4. Calculate 0.372