Geometry & Statistics
Random Sampling and Probability Models
Grade 7
- ✓By the end of this lesson students will be able to understand and identify random samples.
- ✓By the end of this lesson students will be able to use data from a random sample to draw inferences about a population.
- ✓By the end of this lesson students will be able to develop a probability model and use it to find probabilities of events.
- ✓By the end of this lesson students will be able to compare probabilities from a model to observed frequencies from experiments.
Key concepts
A random sample is a subset of a population selected in such a way that every member of the population has an equal chance of being chosen. This method helps ensure the sample is representative of the larger population, allowing for valid inferences.
The entire group of individuals, objects, or data points that a researcher is interested in studying or drawing conclusions about.
In statistics, an inference is a conclusion or generalization made about a population based on data collected from a sample. Random samples are crucial for making reliable inferences.
A probability model describes a random phenomenon by listing all possible outcomes (the sample space) and assigning a probability to each outcome or event. It can be theoretical (based on reasoning) or experimental (based on observed data).
The set of all possible outcomes of a probability experiment. For example, when rolling a standard number cube, the sample space is {1, 2, 3, 4, 5, 6}.
An event is a specific outcome or a collection of outcomes from a probability experiment. For example, when rolling a number cube, 'rolling an even number' is an event that includes the outcomes {2, 4, 6}.
The probability of an event based on reasoning and mathematical calculations, assuming all outcomes are equally likely. It represents what should happen in an ideal situation.
The probability of an event based on the results of an actual experiment or observation. It represents what did happen during a series of trials.
Key facts to remember
- 1A random sample is crucial for making valid and unbiased inferences about a population.
- 2The sum of the probabilities of all possible outcomes in a probability model (sample space) must always equal 1.
- 3Probability values are always between 0 and 1, inclusive (0 for an impossible event, 1 for a certain event).
- 4Theoretical probability represents what is expected to happen, while experimental probability represents what actually happened during an experiment.
- 5As the number of trials in an experiment increases, the experimental probability tends to get closer to the theoretical probability (Law of Large Numbers).
Worked examples
Example 1
A middle school wants to survey its 800 students about their favorite after-school activity. Which method would produce a random sample?
Answer
Option B: Surveying 100 students chosen by assigning each student a number and then using a random number generator to select 100 numbers.
A truly random sample helps ensure the survey results are representative of the entire student body.
Example 2
A spinner is divided into 5 equally sized sections labeled A, B, C, D, and E. It is spun once. Create a theoretical probability model for this experiment and find the probability of landing on a vowel.
Answer
Theoretical Probability Model: P(A)=1/5, P(B)=1/5, P(C)=1/5, P(D)=1/5, P(E)=1/5. The probability of landing on a vowel is 2/5.
The sum of all probabilities in the model (1/5 + 1/5 + 1/5 + 1/5 + 1/5) should always equal 1.
Example 3
A coin is flipped 40 times. It lands on heads 22 times and tails 18 times. \na) Create an experimental probability model for this experiment. \nb) What is the theoretical probability of landing on heads? \nc) Compare the experimental probability of landing on heads to its theoretical probability.
Answer
a) Experimental Probability Model: P(Heads) = 11/20, P(Tails) = 9/20.\nb) Theoretical P(Heads) = 1/2.\nc) The experimental probability of landing on heads (11/20 or 0.55) is slightly higher than the theoretical probability (1/2 or 0.50).
Experimental probability often varies from theoretical probability, especially with a small number of trials. As the number of trials increases, experimental probability usually gets closer to theoretical probability.
Common mistakes
- ✗Confusing a biased sample (e.g., convenience sample) with a random sample, leading to invalid inferences.
- ✗Incorrectly identifying the sample space or miscounting the total number of possible outcomes for an event.
- ✗Expressing probability as a number greater than 1 or less than 0, which is mathematically impossible.
- ✗Assuming that experimental probability will always exactly match theoretical probability, especially with a small number of trials.
- ✗Not simplifying fractions when expressing probabilities, which is standard practice in math.
Exam tips
- ★When asked to identify a random sample, always explain why the chosen method ensures every member of the population has an equal chance of selection.
- ★For probability questions, clearly list the sample space and the favorable outcomes for the event before calculating the probability.
- ★Pay close attention to whether the question asks for theoretical probability (what should happen) or experimental probability (what did happen).
- ★Always simplify probability fractions to their lowest terms unless otherwise specified.
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