VCE Maths Methods Binomial Probability Mini Test 2
Number of marks: 8
Reading time: 1 minute
Writing time: 12 minutes
Section A – Calculator Allowed
Instructions
• Answer all questions in pencil on your Multiple-Choice Answer Sheet.
• Choose the response that is correct for the question.
• A correct answer scores 1; an incorrect answer scores 0.
• Marks will not be deducted for incorrect answers.
• No marks will be given if more than one answer is completed for any question.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Let \(X\) be a discrete random variable with binomial distribution \(X \sim \text{Bi}(n, p)\). The mean and the standard deviation of this distribution are equal.
Given that \(0 < p < 1\), the smallest number of trials, \(n\), such that \(p \le 0.01\) is
- A. 37
- B. 49
- C. 98
- D. 99
- E. 101
Inside a container there are one million coloured building blocks. It is known that 20% of the blocks are red. A sample of 16 blocks is taken from the container. For samples of 16 blocks, \(\hat{P}\) is the random variable of the distribution of sample proportions of red blocks. (Do not use a normal approximation.)
\(\Pr(\hat{P} \ge \frac{3}{16})\) is closest to
- A. 0.6482
- B. 0.8593
- C. 0.7543
- D. 0.6542
- E. 0.3211
A box contains \( n \) green balls and \( m \) red balls. A ball is selected at random, and its colour is noted. The ball is then replaced in the box. In 8 such selections, where \( n \ne m \), what is the probability that a green ball is selected at least once?
- A. \( 8 \left( \frac{n}{n + m} \right) \left( \frac{m}{n + m} \right)^7 \)
- B. \( 1 - \left( \frac{n}{n + m} \right)^8 \)
- C. \( 1 - \left( \frac{m}{n + m} \right)^8 \)
- D. \( 1 - \left( \frac{n}{n + m} \right) \left( \frac{m}{n + m} \right)^7 \)
- E. \( 1 - 8 \left( \frac{n}{n + m} \right) \left( \frac{m}{n + m} \right)^7 \)
The probability of winning a game is 0.25
The probability of winning a game is independent of winning any other game.
If Ben plays 10 games, the probability that he will win exactly four times is closest to
- A. 0.1460
- B. 0.2241
- C. 0.9219
- D. 0.0781
- E. 0.7759
End of Section A
Section B – No Calculator
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
For a certain population the probability of a person being born with the specific gene SPGE1 is \(\frac{3}{5}\).
The probability of a person having this gene is independent of any other person in the population having this gene.
a. In a randomly selected group of four people, what is the probability that three or more people have the SPGE1 gene? 2 marks
b. In a randomly selected group of four people, what is the probability that exactly two people have the SPGE1 gene, given that at least one of those people has the SPGE1 gene? Express your answer in the form \(\frac{a^3}{b^4 - c^4}\), where \(a, b, c \in Z^+\). 2 marks
End of examination questions
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