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Questions and Answers
In a random individual with skin cancer, what is the probability that their cancer is located on their limbs?
In a random individual with skin cancer, what is the probability that their cancer is located on their limbs?
- 0.41
- 0.44 (correct)
- 0.80
- 0.15
What is the probability that a randomly chosen individual with skin cancer on their head is a woman?
What is the probability that a randomly chosen individual with skin cancer on their head is a woman?
- 0.37
- 0.56 (correct)
- 0.15
- 0.44
What is the formula for Bayes' Rule?
What is the formula for Bayes' Rule?
- P(A|B) = P(B|A) / P(A)
- P(A|B) = P(A) * P(B|A) / P(B) (correct)
- P(A|B) = P(B) / (P(A) * P(B|A))
- P(A|B) = P(B) * P(A|B) / P(A)
What is the term used to describe the probability that a person actually has a disease given a positive test result?
What is the term used to describe the probability that a person actually has a disease given a positive test result?
What is the probability that a person with skin cancer will have the disease on their trunk?
What is the probability that a person with skin cancer will have the disease on their trunk?
What is the probability of getting a heads result on a coin toss?
What is the probability of getting a heads result on a coin toss?
Bayes’s theorem states that the probability of an event Ai given another event B is equal to:
Bayes’s theorem states that the probability of an event Ai given another event B is equal to:
Calculate the probability that someone will get lung cancer given that they don't smoke.
Calculate the probability that someone will get lung cancer given that they don't smoke.
Calculate the probability that someone is a smoker and has lung cancer.
Calculate the probability that someone is a smoker and has lung cancer.
Are the events 'Smoker' and 'Lung cancer' independent?
Are the events 'Smoker' and 'Lung cancer' independent?
What formula should be used to calculate the probability of two events A and B both occurring?
What formula should be used to calculate the probability of two events A and B both occurring?
What is the probability of randomly catching two female frogs in a row from the artificial pond?
What is the probability of randomly catching two female frogs in a row from the artificial pond?
Which of these is a correct interpretation of the formula P(A or B) = P(A) + P(B) - P(A and B)?
Which of these is a correct interpretation of the formula P(A or B) = P(A) + P(B) - P(A and B)?
Which of the following is NOT a true statement about independent events?
Which of the following is NOT a true statement about independent events?
If P(A) = 0.4 and P(B) = 0.5, and A and B are independent events, then what is P(A and B)?
If P(A) = 0.4 and P(B) = 0.5, and A and B are independent events, then what is P(A and B)?
Which statement best defines independent events in probability?
Which statement best defines independent events in probability?
In a survey with 50 frogs, where there are 25 males and 25 females, what is the probability of picking a male frog first and then a male frog second when sampling without replacement?
In a survey with 50 frogs, where there are 25 males and 25 females, what is the probability of picking a male frog first and then a male frog second when sampling without replacement?
What is the primary concept illustrated by Bayes’s theorem in probability?
What is the primary concept illustrated by Bayes’s theorem in probability?
Which situation exemplifies dependent events?
Which situation exemplifies dependent events?
What does conditional probability measure?
What does conditional probability measure?
Flashcards
Independent Events
Independent Events
Two events are independent if the occurrence of one does not affect the probability of the other.
Conditional Probability
Conditional Probability
The probability of an event occurring given that another event has occurred.
Sampling without Replacement
Sampling without Replacement
A sampling method where selected individuals are not returned to the population, affecting subsequent probabilities.
General Addition Rule
General Addition Rule
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Bayes’s Theorem
Bayes’s Theorem
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Probability of type O
Probability of type O
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Tree Diagrams
Tree Diagrams
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P(woman | head)
P(woman | head)
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Positive Predictive Value (PPV)
Positive Predictive Value (PPV)
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Bayes' Rule
Bayes' Rule
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Disjoint Events
Disjoint Events
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Multiplication Rule
Multiplication Rule
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Multiplication Rule for Independents
Multiplication Rule for Independents
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Probability Example: Dalmatian
Probability Example: Dalmatian
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Smoker and Lung Cancer
Smoker and Lung Cancer
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Study Notes
Course Information
- Course Title: BMS 511 Biostats & Statistical Analysis
- Chapter: 10
- Topic: Independence and Conditional Probabilities
Previous Learning Objectives
- Probability concepts
- Randomness and probability
- Probability models
- Probability rules
- Discrete vs. continuous models
- Random variables
- Mean and variances for discrete models
- Risk and odds
Learning Objectives
- Define general rules of probability
- Independent events
- Conditional probability
- General addition rule
- Multiplication rule
- Tree diagrams
- Bayes's theorem
- Diagnosis test
Independent Events
- Two events are independent if one event's outcome doesn't change the other's probability
- Examples:
- "Male" and "getting heads when flipping a coin" are independent
- "Male" and "taller than 6 ft" are NOT independent
- "Male" and "high cholesterol" are likely independent
- "Male" and "pregnant" are NOT independent
Sampling without Replacement
- Picking one item from a group and not putting it back affects the next selection's probability
- Successive picks are NOT independent in smaller groups
- Successive picks are "nearly" independent in larger groups (e.g., thousands of frogs)
Conditional Probability
- Conditional probability of event B, given event A: (P(A and B)) / (P(A))
- If A and B are independent: P(B | A) = P(B)
Independence Example
- Example shows probabilities of hearing impairment and blue eyes in Dalmatians
- P(HI and B) = 0.05
- P(HI) = 0.28
- P(B) = 0.11
- P(HI | B) ≈ 0.45
- Events are not independent
Another Independence Example
- (1 of 2) 11% of the population smokes,
- probability a smoker gets lung cancer: 0.34,
- probability a non-smoker gets lung cancer is 0.03.
- (2 of 2) Determine if smoking and lung cancer are independent.
General Addition Rule
- (1 of 2) Addition rule for disjoint events: P(A or B) = P(A) + P(B)
- (2 of 2) General addition rule for any two events A and B: P(A or B) = P(A) + P(B) − P(A and B)
Multiplication Rule
- General multiplication rule: P(A and B) = P(A)P(B|A)
- Multiplication rule for independent events: P(A and B) = P(A)P(B)
Multiplication Rule Examples
- Example of frogs in an artificial pond
- Example of unrelated blood donors
Tree Diagrams
- Used to visually represent probabilities and facilitate calculations
- Example: Skin cancer probabilities among men and women based on body locations
Tree Diagram Example - Diagnostic Tests
- Shows disease rate, sensitivity, specificity
- Explains positive predictive value (PPV)
Bayes' Theorem
- Formula: (P(A|B) = (P(A) * P(B|A)) / P(B))
- Example: Probability of rolling a 3 and 4 on a die.
Bayes's theorem
- A¡, A2, ..., Ak are disjoint events with probabilities that add to 1
- B is any other event
- P(A¡|B) =((P(B|A¡) * P(A¡)) / Σ (P(B|Ai) * P(Ai)))
Diagnosis Tests
- Includes prevalence, sensitivity, specificity
- Shows calculation of probability of having a disease given a positive test result (positive predictive value)
Diagnostic Tests Examples: HIV-AIDS
- Example using an enzyme immunoassay test.
Another Diagnostic Tests Example (prostate cancer)
- Prostate cancer rates
- PSA test sensitivity and specificity
- Calculating positive predictive value
Comparison of Screening/Diagnostic Test Results with Actual Disease Status
- Summarizes test results for disease vs. no disease.
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