In a certain population, 22% of people are smokers, 57% of people are males and 12% males are smokers. If a person is chosen at random from the population, what is that probality that selected person is either a male or a smoker?
0.67
Understanding Probability of Events
The question asks for the probability that a randomly selected person from a population is either a male or a smoker. This involves calculating the probability of the union of two events: being a male and being a smoker. We are given the individual probabilities of these events and the probability of a male being a smoker, which we will interpret as the probability of a person being both male and a smoker.
Given Probabilities
We are provided with the following information:
- Probability of a person being a smoker: P(Smoker) = 22% = 0.22
- Probability of a person being male: P(Male) = 57% = 0.57
- Probability that 12% males are smokers: We interpret this as the probability of a person being both male AND a smoker: P(Male and Smoker) = 12% = 0.12
Calculating Probability of Male or Smoker
To find the probability that the selected person is either a male or a smoker, we use the formula for the probability of the union of two events A and B:
\(\text{P(A or B)} = \text{P(A)} + \text{P(B)} - \text{P(A and B)}\)
In this case, Event A is selecting a male, and Event B is selecting a smoker. So, we want to find P(Male or Smoker).
Using the formula and the given probabilities:
\(\text{P(Male or Smoker)} = \text{P(Male)} + \text{P(Smoker)} - \text{P(Male and Smoker)}\)
\(\text{P(Male or Smoker)} = 0.57 + 0.22 - 0.12\)
\(\text{P(Male or Smoker)} = 0.79 - 0.12\)
\(\text{P(Male or Smoker)} = 0.67\)
Thus, the probability that the selected person is either a male or a smoker is 0.67.
Summary of Calculation Steps
- Identify the events: Event A = Male, Event B = Smoker.
- Note the given probabilities: P(Male) = 0.57, P(Smoker) = 0.22, P(Male and Smoker) = 0.12.
- Apply the formula for the union of two events: P(Male or Smoker) = P(Male) + P(Smoker) - P(Male and Smoker).
- Substitute the values: P(Male or Smoker) = 0.57 + 0.22 - 0.12.
- Calculate the result: P(Male or Smoker) = 0.67.
| Event | Probability |
|---|---|
| Smoker (S) | 0.22 |
| Male (M) | 0.57 |
| Male and Smoker (M and S) | 0.12 |
| Male or Smoker (M or S) | P(M) + P(S) - P(M and S) = 0.57 + 0.22 - 0.12 = 0.67 |
Revision Table: Key Probability Formulas
| Concept | Formula | Explanation |
|---|---|---|
| Probability of A or B (Union) | \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) | Used when finding the probability that at least one of two events occurs. Subtracts the intersection to avoid double-counting. |
| Probability of A and B (Intersection - Independent Events) | \(P(A \cap B) = P(A) \times P(B)\) | Used if events A and B are independent. |
| Probability of A and B (Intersection - Dependent Events) | \(P(A \cap B) = P(A|B) \times P(B) = P(B|A) \times P(A)\) | Used if events A and B are dependent. P(A|B) is the probability of A given B. |
| Conditional Probability | \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) | The probability of event A occurring given that event B has already occurred. |
Additional Information: Probability Concepts
Probability is a measure of the likelihood that an event will occur. It is a value between 0 and 1, inclusive.
- Event: A specific outcome or a set of outcomes.
- Sample Space: The set of all possible outcomes of a random experiment.
- Union of Events (\(A \cup B\)): The event that occurs if A occurs, or B occurs, or both occur. Represented as "A or B".
- Intersection of Events (\(A \cap B\)): The event that occurs if both A and B occur. Represented as "A and B".
- Mutually Exclusive Events: Two events that cannot occur at the same time. If A and B are mutually exclusive, \(P(A \cap B) = 0\), and \(P(A \cup B) = P(A) + P(B)\).
- Independent Events: Two events where the occurrence of one does not affect the probability of the other. If A and B are independent, \(P(A \cap B) = P(A) \times P(B)\) and \(P(A|B) = P(A)\).
In this problem, being male and being a smoker are not mutually exclusive (as some males are smokers) and not independent (as the probability of being a smoker might be different for males and females, indicated by the provided P(Male and Smoker)). Therefore, the general formula for the union must be used.
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