A binomial distribution counts successes in n independent trials, and for B(n, p) the mean is np, so this IB sheet opens with the distribution's object and its mean. The variance np(1 − p) and the normal curve's symmetry appear in the answers.
Key Takeaways
A binomial distribution counts successes in n independent trials (Q1).
The mean of B(n, p) is np (Q2).
Q3 records the variance as np(1 − p).
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.A binomial distribution counts the number of
(a) successes in n independent trials(b) failures in a single trial(c) values in a data set(d) errors in a sample
2.For a binomial distribution B(n, p), the mean is
(a) np(b) n + p(c) n ÷ p(d) p
3.The variance of a binomial distribution B(n, p) is
(a) np(1 − p)(b) np(c) n(1 − p)(d) p(1 − p)
4.The normal distribution is
(a) symmetric about the mean(b) skewed to the right(c) skewed to the left(d) always flat
5.The mean of a normal distribution is denoted by
(a) μ(b) σ(c) x̄(d) n
6.In a normal distribution, about 68% of the values lie within
(a) 1 standard deviation of the mean(b) 2 standard deviations of the mean(c) 3 standard deviations of the mean(d) half a standard deviation of the mean
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.State the conditions needed for a binomial distribution.
8.Write the mean and variance of a binomial distribution.
9.What shape does the normal distribution have?
10.Write the notation for a normal distribution with mean μ and variance σ².
11.About what percentage of values lie within 2 standard deviations of the mean in a normal distribution?
12.Define the standard deviation.
13.What symbol is used for the standard deviation?
14.Give one example of data that might follow a normal distribution.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.For X ~ B(10, 0.5), find the mean using np.
16.For X ~ B(20, 0.25), find the mean and the variance.
17.A fair coin is tossed 8 times. Using the binomial mean np with p = 0.5, find the expected number of heads.
18.For X ~ N(50, 100), state the mean and the standard deviation.
19.For X ~ N(100, 25), about 68% of the values lie within which interval?
20.A fair die is rolled 12 times. Using the binomial mean np with p = 1 ÷ 6, find the expected number of sixes.
Answer Key
1. a) successes in n independent trials
2. a) np
3. a) np(1 − p)
4. a) symmetric about the mean
5. a) μ
6. a) 1 standard deviation of the mean
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What does a binomial distribution count?
Successes in n independent trials, option (a) in this worksheet (Q1).
What is the mean of a binomial distribution B(n, p)?
np, option (a) in this worksheet (Q2).
What is the variance of B(n, p)?
np(1 − p), option (a) in this worksheet (Q3).
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