For X ~ B(10, 0.4) the mean is E(X) = 4 and for X ~ B(10, 0.5) the variance is Var(X) = 2.5, so this Year 13 sheet opens with the binomial's mean and variance. Conditional probability and a 0.125 answer appear in the key.
Key Takeaways
E(X) = 4 for X ~ B(10, 0.4) (Q1).
Var(X) = 2.5 for X ~ B(10, 0.5) (Q2).
Q5 records P(A ∩ B) ÷ P(B).
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.If X ~ B(10, 0.4), then E(X) =
(a) 4(b) 2.4(c) 6(d) 0.4
2.If X ~ B(10, 0.5), then Var(X) =
(a) 2.5(b) 5(c) 10(d) 0.5
3.If X ~ B(3, 0.5), then P(X = 0) =
(a) 0.125(b) 0.5(c) 0.375(d) 0.25
4.If Z ~ N(0, 1), then P(Z < 0) =
(a) 0.5(b) 0(c) 1(d) 0.95
5.The conditional probability P(A | B) is equal to:
(a) P(A ∩ B) ÷ P(B)(b) P(A ∩ B) ÷ P(A)(c) P(A) × P(B)(d) P(A ∪ B) ÷ P(B)
6.The mean of the data set 2, 4, 6, 8, 10 is:
(a) 6(b) 5(c) 30(d) 4
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.If X ~ B(10, 0.5), find P(X = 10).
8.If X ~ B(5, 0.2), find P(X = 0).
9.Find the mean and the variance of the data set 1, 2, 3, 4, 5.
10.If Z ~ N(0, 1), state the value of P(Z ≤ 1.96) to three significant figures.
11.Given P(A ∩ B) = 0.2 and P(B) = 0.4, find P(A | B).
12.If X ~ B(4, 0.5), find P(X = 1).
13.For a binomial distribution X ~ B(n, p), write down the formulas for E(X) and Var(X).
14.A fair six-sided die is rolled twice. Find the probability that the sum of the two scores is 2.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.A fair coin is tossed 6 times. Find the probability of getting 6 heads.
16.The weight of a bag of flour is normally distributed with mean 1000 g and standard deviation 40 g. Find the probability that a randomly chosen bag weighs less than 1000 g.
17.A multiple-choice test has 10 questions, each with 4 options. A student guesses every answer. Find the probability that all 10 answers are correct.
18.In a factory, 5% of items are defective. A random sample of 20 items is taken. Using X ~ B(20, 0.05), find P(X = 0), giving your answer to two significant figures.
19.A biased coin has P(heads) = 0.6. It is tossed 3 times. Find the probability of getting exactly two heads.
20.The heights of plants are normally distributed with mean 50 cm and standard deviation 10 cm. Find the probability that a randomly chosen plant is taller than 50 cm.
Answer Key
1. a) 4
2. a) 2.5
3. a) 0.125
4. a) 0.5
5. a) P(A ∩ B) ÷ P(B)
6. a) 6
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 is E(X) for X ~ B(10, 0.4)?
4, option (a) in this worksheet (Q1).
What is Var(X) for X ~ B(10, 0.5)?
2.5, option (a) in this worksheet (Q2).
What is the formula for conditional probability?
P(A ∩ B) ÷ P(B), option (a) in this worksheet (Q5).
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