Question

unit 4: probability, random variables, and probability distributions
hw 8 – the geometric distribution name: lucy luo

1. for the following situations, decide if it is a binomial setting, a geometric setting, or neither. explain your answer.

a) you keep drawing cards out a deck, without replacement, until an ace is drawn
two possible outcomes : ace/not ace
fixed probability :
independent trials : ×
first success : √
neither
b) you roll a dice 20 times and record the number of sixes you have rolled.
two outcomes : six/not six
fixed probability
independent trials : √
c) crest claims that 40% of americans use their toothpaste. you take a random sample of 50 americans and count how many use crest toothpaste.
two outcomes : use toothpaste/not use
fixed probability :.40
d) you flip a coin until you get tails.
two outcomes : head/tail
fixed probability :.50
independent trials : √
e) 5% of the tomatoes at a farmer’s market have imperfections on them. you randomly choose one tomato at a time until you find one with an imperfection.
two outcomes : imperfection/perfection
fixed probability :.05
independent trials : √
first success : √
geometric
f) 5% of the tomatoes at the farmer’s market have imperfections on them. you randomly choose 20 tomatoes and count the number of imperfections on them.
two outcomes : imperfection/perf
fixed probability :.05
independent trials : √

Explanation:

• a) In a binomial setting, trials are independent. Here, cards are drawn without replacement, so the probability changes with each draw (trials are not independent). In a geometric setting, we also need independent trials. So this is neither.
• b) There are a fixed number of trials (\(n = 20\)), two possible outcomes (six or not six), and the probability of getting a six (\(p=\frac{1}{6}\)) is fixed for each independent roll. This is a binomial setting.
• c) There are a fixed number of trials (\(n = 50\)), two possible outcomes (use Crest or not), and a fixed probability (\(p = 0.4\)) for each independent trial (assuming the sample is random). This is a binomial setting.
• d) There are two possible outcomes (head or tail), a fixed probability (\(p=0.5\)) for each independent flip, and we are looking for the first success (getting a tail). This is a geometric setting.
• e) There are two possible outcomes (imperfection or no imperfection), a fixed probability (\(p = 0.05\)) for each independent choice (assuming the market has a large number of tomatoes so that the probability is approximately fixed), and we are looking for the first success (finding a tomato with an imperfection). This is a geometric setting.
• f) There are a fixed number of trials (\(n=20\)), two possible outcomes (imperfection or no imperfection), and a fixed probability (\(p = 0.05\)) for each independent trial (assuming the market has a large number of tomatoes so that the probability is approximately fixed). This is a binomial setting.

Answer:

a) Neither.
b) Binomial setting.
c) Binomial setting.
d) Geometric setting.
e) Geometric setting.
f) Binomial setting.