📝 Chapter 5: Discrete Probability Distributions

For \( X \sim \mathrm{Binomial}(15, 0.4) \), find \( E(X) \).

A coin is tossed 12 times. Find \( P(\text{exactly 8 heads}) \). (decimal to 4 places)

A free-throw shooter has 90% accuracy. \( P(\text{first miss is on 5th attempt}) \) = ? (decimal to 4 places)

A bag has 12 marbles: 4 red, 8 blue. 3 are drawn without replacement. \( P(\text{exactly 2 red}) \) = ? (decimal to 4 places)

For a binomial distribution with \( n = 25 \) and \( p = 0.4 \), find \( \sigma \) (standard deviation, decimal to 4 places).

A salesperson succeeds on 25% of calls. The expected number of calls until the first sale is:

A 30-question test, 5 options each. A student guesses on every question. Expected number correct:

A factory's quality control: 95% of items pass inspection. From a batch of 200 items, what is the probability that at least 195 pass? Show all work using the binomial distribution.

A box contains 20 chocolates: 6 with caramel, 14 plain. A sample of 4 is drawn at random. Construct the probability distribution for \( X \) = number of caramel chocolates. Find \( E(X) \) and \( \mathrm{Var}(X) \).

A baseball player has a batting average of .250. Find the probability his first hit comes on his 5th at-bat. What is the most probable at-bat for the first hit? Justify.

For each scenario, identify the distribution (binomial, geometric, hypergeometric) and justify:
a) Drawing 3 cards from a 52-card deck and counting Aces.
b) Tossing a coin until the first head.
c) Asking 100 randomly chosen voters a yes/no question.

Explain the difference between binomial and hypergeometric distributions. Use a concrete example (e.g., a deck of cards) to demonstrate when each applies. What is the role of "with vs. without replacement"?

A student claims: "If a coin is tossed 10 times, the most likely number of heads is exactly 5." Is this correct? Justify with a calculation. What is \( P(X=5) \) for \( X \sim \mathrm{Binomial}(10, 0.5) \)?

Describe a step-by-step procedure for deciding whether a problem is binomial, geometric, or hypergeometric. Apply your procedure to: "Drawing 5 cards from a deck and counting Hearts."

Compare the means and variances: binomial \( np, np(1-p) \); geometric \( \dfrac{1}{p} \); hypergeometric \( n\dfrac{r}{N} \). What does the variance tell us about the spread of outcomes? Give an example.

A pharmaceutical drug is effective in 70% of patients. In a clinical trial of 25 patients, find the probability that exactly 20 are cured. Find the expected number cured and the standard deviation.

A quality control inspector takes a sample of 5 items from a batch of 50. The batch contains 3 defective items. a) Find \( P(\text{at least 1 defective in sample}) \). b) The inspector rejects the batch if any defective item is found. \( P(\text{batch is rejected}) \) = ?

A telemarketer calls until she makes her first sale. Each call has a 0.15 probability of success. a) Find the probability she succeeds on the 6th call. b) What is the expected number of calls until the first sale?

A company manufactures microchips with a 2% defect rate. Quality assurance tests samples of 100 chips. a) Find the expected number of defective chips per sample. b) Find \( P(\text{at most 1 defective}) \) using the binomial distribution. c) If a sample contains 5 or more defectives, the entire batch is destroyed. Estimate \( P(\text{destruction}) \).