Compiled by: Charles Wanjohi
Calculation of probability is fundamental when making decisions based on data in community health. Health professionals in Kenya often rely on probabilities to assess risks, predict disease outbreaks, and evaluate treatment outcomes. Understanding how to calculate probability helps in interpreting health surveys, clinical trial results, and epidemiological data. This chapter focuses on practical applications of probability in health care delivery using worked examples relevant to community health settings.
Probability measures the likelihood of an event occurring, expressed as a value between 0 and 1. An event with probability 0 is impossible, while 1 indicates certainty. The formula for probability of an event \( A \) is:
$$ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} $$
Example 1: A health worker randomly selects a patient from a clinic with 100 patients, 30 of whom have malaria. What is the probability that the selected patient has malaria?
Given:
Number of favorable outcomes = 30
Total outcomes = 100
$$ P(\text{malaria}) = \frac{30}{100} $$
$$ = 0.3 $$
Answer: 0.3 (30%)
Example 2: In a sample of 120 children, 48 are vaccinated against measles. What is the probability a randomly selected child is not vaccinated?
Given:
Number vaccinated = 48
Total children = 120
Number not vaccinated = 120 - 48 = 72
$$ P(\text{not vaccinated}) = \frac{72}{120} $$
$$ = 0.6 $$
Answer: 0.6 (60%)
Example 3: A community health survey shows that out of 500 households, 125 use clean water sources. What is the probability that a household uses unclean water?
Given:
Households using clean water = 125
Total households = 500
Households using unclean water = 500 - 125 = 375
$$ P(\text{unclean water}) = \frac{375}{500} $$
$$ = 0.75 $$
Answer: 0.75 (75%)
Example 4: In a hospital, the probability that a patient has hypertension is 0.4. What is the probability that a patient selected at random does not have hypertension?
Given:
\( P(\text{hypertension}) = 0.4 \)
$$ P(\text{no hypertension}) = 1 - P(\text{hypertension}) $$
$$ = 1 - 0.4 $$
$$ = 0.6 $$
Answer: 0.6 (60%)
Example 5: A health facility has 200 patients, 50 have diabetes, 30 have tuberculosis, and the rest have other illnesses. What is the probability that a randomly selected patient has diabetes or tuberculosis?
Given:
Patients with diabetes = 50
Patients with tuberculosis = 30
Total patients = 200
$$ P(\text{diabetes or tuberculosis}) = \frac{50 + 30}{200} $$
$$ = \frac{80}{200} $$
$$ = 0.4 $$
Answer: 0.4 (40%)
Two events are independent if the occurrence of one does not affect the probability of the other. The probability of both independent events \( A \) and \( B \) occurring is:
$$ P(A \text{ and } B) = P(A) \times P(B) $$
Example 1: The probability that a patient has malaria is 0.2, and the probability that the same patient has typhoid is 0.1. Assuming these are independent, what is the probability that the patient has both malaria and typhoid?
Given:
\( P(\text{malaria}) = 0.2 \)
\( P(\text{typhoid}) = 0.1 \)
$$ P(\text{malaria and typhoid}) = 0.2 \times 0.1 $$
$$ = 0.02 $$
Answer: 0.02 (2%)
Example 2: In a health screening, the probability of detecting high blood pressure is 0.3, and the probability of detecting high blood sugar is 0.25. What is the probability that a patient has both conditions?
Given:
\( P(\text{high blood pressure}) = 0.3 \)
\( P(\text{high blood sugar}) = 0.25 \)
$$ P(\text{both}) = 0.3 \times 0.25 $$
$$ = 0.075 $$
Answer: 0.075 (7.5%)
Example 3: The probability that a patient receives vaccination A is 0.7, and vaccination B is 0.8. What is the probability that a patient receives both vaccines?
Given:
\( P(A) = 0.7 \)
\( P(B) = 0.8 \)
$$ P(A \text{ and } B) = 0.7 \times 0.8 $$
$$ = 0.56 $$
Answer: 0.56 (56%)
Example 4: If the probability of a health worker attending a training is 0.6 and the probability of the same worker receiving a promotion is 0.2, what is the probability that both happen?
Given:
\( P(\text{training}) = 0.6 \)
\( P(\text{promotion}) = 0.2 \)
$$ P(\text{training and promotion}) = 0.6 \times 0.2 $$
$$ = 0.12 $$
Answer: 0.12 (12%)
Example 5: The probability that a randomly selected household uses mosquito nets is 0.9, and the probability that the household has a child under five is 0.3. Assuming independence, what is the probability the household uses mosquito nets and has a child under five?
Given:
\( P(\text{mosquito nets}) = 0.9 \)
\( P(\text{child under five}) = 0.3 \)
$$ P(\text{both}) = 0.9 \times 0.3 $$
$$ = 0.27 $$
Answer: 0.27 (27%)
Mutually exclusive events cannot occur at the same time. The probability of either event \( A \) or event \( B \) occurring is:
$$ P(A \text{ or } B) = P(A) + P(B) $$
Example 1: In a health camp, the probability that a patient has malaria is 0.3, and the probability that the patient has typhoid is 0.2. What is the probability that the patient has either malaria or typhoid, assuming the conditions are mutually exclusive?
Given:
\( P(\text{malaria}) = 0.3 \)
\( P(\text{typhoid}) = 0.2 \)
$$ P(\text{malaria or typhoid}) = 0.3 + 0.2 $$
$$ = 0.5 $$
Answer: 0.5 (50%)
Example 2: A patient can be diagnosed with either pneumonia or bronchitis, but not both. The probabilities are 0.4 and 0.1 respectively. What is the probability of having either disease?
Given:
\( P(\text{pneumonia}) = 0.4 \)
\( P(\text{bronchitis}) = 0.1 \)
$$ P(\text{either}) = 0.4 + 0.1 $$
$$ = 0.5 $$
Answer: 0.5 (50%)
Example 3: In a clinic, the probability of a patient having flu is 0.25, and the probability of having a cold is 0.35. These are mutually exclusive. Find the probability of having flu or cold.
Given:
\( P(\text{flu}) = 0.25 \)
\( P(\text{cold}) = 0.35 \)
$$ P(\text{flu or cold}) = 0.25 + 0.35 $$
$$ = 0.6 $$
Answer: 0.6 (60%)
Example 4: The probability of a patient being diagnosed with diabetes is 0.15, and the probability of being diagnosed with hypertension is 0.25. These are mutually exclusive events. Calculate the probability of either diagnosis.
Given:
\( P(\text{diabetes}) = 0.15 \)
\( P(\text{hypertension}) = 0.25 \)
$$ P(\text{diabetes or hypertension}) = 0.15 + 0.25 $$
$$ = 0.4 $$
Answer: 0.4 (40%)
Example 5: At a health facility, the probability that a patient is tested positive for HIV is 0.1, and the probability that the patient is tested positive for hepatitis B is 0.05. These are mutually exclusive. What is the probability that the patient tests positive for either disease?
Given:
\( P(\text{HIV}) = 0.1 \)
\( P(\text{hepatitis B}) = 0.05 \)
$$ P(\text{HIV or hepatitis B}) = 0.1 + 0.05 $$
$$ = 0.15 $$
Answer: 0.15 (15%)
When two events can occur simultaneously, the probability of either event \( A \) or \( B \) occurring is:
$$ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $$
This accounts for the overlap where both events happen.
Example 1: In a community, the probability that a person has malaria is 0.3, the probability of having typhoid is 0.2, and the probability of having both is 0.05. Calculate the probability that a person has malaria or typhoid.
Given:
\( P(\text{malaria}) = 0.3 \)
\( P(\text{typhoid}) = 0.2 \)
\( P(\text{both}) = 0.05 \)
$$ P(\text{malaria or typhoid}) = 0.3 + 0.2 - 0.05 $$
$$ = 0.45 $$
Answer: 0.45 (45%)
Example 2: The probability that a patient has high blood pressure is 0.4, the probability of diabetes is 0.3, and the probability of both conditions is 0.1. Find the probability the patient has high blood pressure or diabetes.
Given:
\( P(\text{hypertension}) = 0.4 \)
\( P(\text{diabetes}) = 0.3 \)
\( P(\text{both}) = 0.1 \)
$$ P(\text{hypertension or diabetes}) = 0.4 + 0.3 - 0.1 $$
$$ = 0.6 $$
Answer: 0.6 (60%)
Example 3: In a screening, the probability of detecting anemia is 0.25, the probability of detecting malaria is 0.35, and the probability of detecting both is 0.1. Calculate the probability of detecting anemia or malaria.
Given:
\( P(\text{anemia}) = 0.25 \)
\( P(\text{malaria}) = 0.35 \)
\( P(\text{both}) = 0.1 \)
$$ P(\text{anemia or malaria}) = 0.25 + 0.35 - 0.1 $$
$$ = 0.5 $$
Answer: 0.5 (50%)
Example 4: The probability of a patient having tuberculosis is 0.15, the probability of HIV is 0.2, and the probability of both is 0.05. What is the probability that a patient has tuberculosis or HIV?
Given:
\( P(\text{tuberculosis}) = 0.15 \)
\( P(\text{HIV}) = 0.2 \)
\( P(\text{both}) = 0.05 \)
$$ P(\text{tuberculosis or HIV}) = 0.15 + 0.2 - 0.05 $$
$$ = 0.3 $$
Answer: 0.3 (30%)
Example 5: In a health survey, the probability that a household uses treated water is 0.7, the probability that the household uses mosquito nets is 0.6, and the probability that the household uses both is 0.5. Calculate the probability that a household uses treated water or mosquito nets.
Given:
\( P(\text{treated water}) = 0.7 \)
\( P(\text{mosquito nets}) = 0.6 \)
\( P(\text{both}) = 0.5 \)
$$ P(\text{treated water or mosquito nets}) = 0.7 + 0.6 - 0.5 $$
$$ = 0.8 $$
Answer: 0.8 (80%)
In a clinic, 40% of patients have hypertension, and 25% have diabetes. If 10% have both, what is the probability that a patient has either hypertension or diabetes? (5 marks)
The probability that a community member uses a latrine is 0.85. What is the probability that a randomly selected member does not use a latrine? (3 marks)
In a survey, the probability of a household owning a bed net is 0.6, and owning a water filter is 0.4. Assuming independence, what is the probability that a household owns both? (5 marks)
Among 150 patients, 45 have malaria and 30 have typhoid. Assuming these are mutually exclusive, what is the probability that a patient has either disease? (4 marks)
The probability that a patient receives treatment A is 0.5, and treatment B is 0.7. If these events are independent, what is the probability that the patient receives neither treatment? (6 marks)
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Create a free accountThis chapter focused on applying inferential statistics to draw conclusions in health care delivery. It began with the calculation of probability, explaining how to determine the likelihood of events occurring within a health context. The chapter then covered the determination of population mean and proportions, showing how to estimate average values and proportions from sample data to make informed decisions. Emphasis was placed on using these statistical tools to interpret health data accurately and support evidence-based practices. The concepts were linked to practical health care scenarios where understanding variability and uncertainty is crucial. Overall, the chapter provided a foundation for using probability and population parameters to enhance health care analysis and decision-making.
In a community health survey, the probability of a patient having malaria is 0.3. What is the probability that a randomly selected patient does not have malaria? (2 marks)
In a sample of 100 patients tested for a disease, 25 tested positive. Calculate the sample proportion of patients who tested positive. (2 marks)