Math Problem Statement
What are the different types of sampling? Explain giving examples. (6) (b) A crucial game of a basketball team depends on the health of their key player. According to his doctor's report, there is a 40% chance that he will be fully fit to play, a 30% chance that he will be somewhat fit to play and a 30% chance that he will not be able to play at all. The coach has estimated the chances of winning at 80% if the player is fully fit, 60% if he is somewhat fit and 40% if he is unable to play.
What the probability that the team will win the game? (ii) If the team has won the game, what is the probability that the key player has been fully fit to play in the game?
Solution
Part (a): Different Types of Sampling with Examples
-
Simple Random Sampling (SRS)
- Definition: Every member of the population has an equal chance of being selected.
- Example: Assigning numbers to all students in a school and using a random number generator to select 50 students for a survey.
-
Stratified Sampling
- Definition: The population is divided into subgroups (strata) based on shared characteristics, and random samples are taken from each subgroup.
- Example: Dividing employees of a company into departments (HR, Finance, IT) and randomly selecting 10 employees from each department.
-
Cluster Sampling
- Definition: The population is divided into clusters, and some clusters are randomly selected. All members of the chosen clusters are surveyed.
- Example: Dividing a city into neighborhoods and randomly selecting 5 neighborhoods to survey every household in them.
-
Systematic Sampling
- Definition: A fixed, periodic interval is used to select members from the population.
- Example: Selecting every 10th customer entering a store for feedback.
-
Convenience Sampling
- Definition: Samples are taken based on ease of access rather than randomness.
- Example: Surveying people at a mall because it is nearby.
-
Quota Sampling
- Definition: The researcher ensures specific subgroups are represented but selects individuals non-randomly until a quota is met.
- Example: Interviewing 20 men and 20 women from a gym without random selection.
Part (b): Calculating Probabilities
(i) Probability of Winning the Game
Let be the event that the team wins the game.
- Probability of the player being fully fit:
- Probability of the player being somewhat fit:
- Probability of the player being unfit:
- Conditional probabilities of winning:
- , ,
Using the law of total probability: Substitute the values:
Thus, the probability that the team will win the game is 0.62 or 62%.
(ii) Probability the Key Player Was Fully Fit Given the Team Won
This is a conditional probability: Using : Substitute values:
Thus, the probability that the key player was fully fit given the team won is approximately 0.516 or 51.6%.
Summary
- Probability of winning the game: 62%
- Probability the player was fully fit given the team won: 51.6%
Do you want detailed explanations of any part or have additional questions?
Related Questions
- What are the advantages and disadvantages of each type of sampling method?
- How is stratified sampling different from cluster sampling in practical applications?
- What are the assumptions in calculating conditional probabilities?
- Can systematic sampling lead to bias? Why or why not?
- How does the law of total probability apply in real-life decision-making?
Tip
When solving probability problems, always clearly define the events and organize the given probabilities to simplify calculations.
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Math Problem Analysis
Mathematical Concepts
Probability
Conditional Probability
Law of Total Probability
Bayes' Theorem
Formulas
Law of Total Probability: P(W) = P(W|F)P(F) + P(W|S)P(S) + P(W|U)P(U)
Conditional Probability: P(F|W) = P(F ∩ W) / P(W)
Intersection Probability: P(F ∩ W) = P(W|F)P(F)
Theorems
Law of Total Probability
Bayes' Theorem
Suitable Grade Level
Grades 11-12
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