Probability Theory Basics

Challenge: Solving the Task Using Bayes' Theorem

Situation Description

Imagine a medical study involving two groups of people:

Group $H$ : 750 individuals with heart problems;
Group $S$ : 800 individuals with chronic stomachache.

We know the following about diabetes prevalence:

Among group $H$ , 7% have diabetes — this is the conditional probability $P(D∣H)=0.07$ , meaning the probability that a person has diabetes ( $D$ ) given they have a heart problem ( $H$ );
Among group $S$ , 12% have diabetes — this is $P(D∣S)=0.12$ , the probability of diabetes given stomachache.

Here, the letters represent:

$H$ : event "person has a heart problem";
$S$ : event "person has a stomachache";
$D$ : event "person has diabetes".

We want to analyze the overall population formed by these two groups combined.

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Calculate $P(H)$ , the probability that a randomly selected person (from both groups combined) has a heart problem.
Calculate $P(S)$ , the probability that a randomly selected person has a stomachache.
Calculate $P(D)$ , the probability that a randomly selected person has diabetes.

Finally, use Bayes’ theorem to calculate the probability that a randomly selected person with diabetes has a chronic stomachache, expressed as:

P(S∣D)= \frac{P(D∣S) \times P(S)}{P(D)}

Løsning

# Probability that ramdomly selected person has heart problem

P_heart = 750 / 1550

# Probability that ramdomly selected person has stomachache

P_stomach = 800 / 1550

P_heart_diabetes = 0.07

P_stomach_diabetes = 0.12

# Probability that you randomly select a person that has diabetes

P_random_diabetes = P_heart * P_heart_diabetes + P_stomach * P_stomach_diabetes

# Randomly selected person with diabetes has chronic stomachache

P_diabetes_belong_stomach = (P_stomach * P_stomach_diabetes)/P_random_diabetes

print(f'The probability is {P_diabetes_belong_stomach:.4f}')

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Sektion 2. Kapitel 6

single

# Probability that randomly selected person has heart problem

P_heart = ___ / 1550

# Probability that randomly selected person has stomachache

P_stomach = ___

P_heart_diabetes = 0.07

P_stomach_diabetes = 0.12

# Probability that you randomly select a person that has diabetes

P_random_diabetes = ___ * P_heart_diabetes + P_stomach * ___

# Randomly selected person with diabetes has chronic stomachache

P_diabetes_belong_stomach = (P_stomach * P_stomach_diabetes)/P_random_diabetes

print(f'The probability is {P_diabetes_belong_stomach:.4f}')

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