4.2: The Geometric Distribution
- Last updated
- Nov 14, 2024
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- 51648
- Hannah Seidler-Wright
- Chaffey College
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There are many probability experiments where a trial has only two outcomes. For example, asking a group of individuals if they vote yes on a proposition, or randomly guessing on a multiple choice test. When we conduct a sequence of independent trials with only two outcomes per trial, we are conducting a binomial experiment.
- Which of the following only have two possible outcomes:
- Rolling a 4 on a 6-sided die
- Examining the global temperature change over time.
- Measuring the height of adult in California
- Meeting a person that is infected with Covid-19
- Rolling a 4 on a 6-sided die
Characteristics of a Geometric Experiment
A geometric experiment is a probability experiment with the following characteristics:
- Each trial has exactly two possible outcomes which are labeled success and failure.
- The probability of success is the same for each trial. We denote the probability of success as p and the probability of failure as q=1−p.
- We look for when the first and only success occurs. There must be at least one trial, and in theory, we could repeat trials forever.
- Go to https://www.random.org/dice/ and roll 1 die. Roll the die counting the number of trials it took to roll a 5. Keep track of your rolls in the table below.
Tally
On what attempt did you succeed in rolling a 5?
- Assume we will roll a fair six-sided die.
- What is the probability of rolling a 5? We define rolling a 5 as success, and therefore, we are computing the probability of success.
- What is the probability that we will not roll a 5? Use the complement rule to compute the probability of failure.
- What is the probability of rolling a 5? We define rolling a 5 as success, and therefore, we are computing the probability of success.
- Suppose we are rolling a fair six-sided die.
- What is the probability that we will roll a 5 (succeed) on the first attempt?
- What is the probability that we will roll a 5 (succeed) on the second attempt? In this case, we fail on the first try and succeed on the second try. Use the multiplication rule for independent events.
- What is the probability that we will roll a 5 (succeed) on the first attempt?
- Suppose we are rolling a fair six-sided die.
- What is the probability that we will roll a die and succeed (roll a 5) on the third attempt? In this case, we fail on the first and second tries and succeed on the third try. Use the multiplication rule for independent events.
- What is the probability that we will roll a die and succeed (roll a 5) on the fourth attempt? In this case, we fail on the first and second and third tries and succeed on the fourth try. Use the multiplication rule for independent events.
- What is the probability that we will roll a die and succeed (roll a 5) on the third attempt? In this case, we fail on the first and second tries and succeed on the third try. Use the multiplication rule for independent events.
- What is the formula for computing geometric probability? So far, this is what we have come up with:
P(1)=16P(2)=56⋅16P(3)=56⋅56⋅16=(56)2⋅16P(4)=56⋅56⋅56⋅16=(56)3⋅16
What patterns do you notice?
- What is P(5)?
Geometric Probability
In general, the probability of succeeding only once on the xth attempt is
P(x)=qx−1p
where p is the probability of success and q=1−p is the probability of failure.
- You try! You enter a darts tournament. The probability of hitting the bullseye is 17%. What is the probability that you hit the bullseye on the 7th attempt? You can upload an image to show your thinking.