Purpose: Confirm prerequisite skills from Ch 1–4: combinations, basic probability, expected value, and discrete probability distributions.
Prerequisite Knowledge from Ch 1–4
Question 1 [3 marks]
Evaluate without a calculator (show steps):
a) \( \binom{6}{2} \cdot (0.5)^6 \)
b) \( \binom{4}{0} \cdot (0.7)^4 \cdot (0.3)^0 \)
c) \( (0.8)^3 \cdot 0.2 \)
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Question 2 [3 marks]
A coin is tossed 3 times. Construct the probability distribution table for \( X \) = number of heads. Verify that \( \sum P = 1 \).
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Question 3 [3 marks]
For the distribution in Q2:
a) Compute \( E(X) \).
b) Compute \( E(X^2) \).
c) Compute \( \mathrm{Var}(X) = E(X^2) - [E(X)]^2 \).
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Question 4 [2 marks]
From a bag with 4 red and 6 blue marbles, 2 are drawn without replacement. Use combinations to find \( P(\text{both red}) \).
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Question 5 [2 marks]
A trial has \( P(\text{success}) = 0.3 \). What does it mean for trials to be "independent"? Give a concrete example involving repeated dice rolls.
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Question 6 [3 marks]
Reflection: As we move into binomial, geometric, and hypergeometric distributions, what concept from earlier chapters do you most need to review? Be specific (e.g., "factorial cancellation," "computing combinations," etc.).