Question

What is the probability that the selected person is a female?

Answer 1

Concept: The classical definition of probability states that the probability of an event \(E\) occurring is given by the ratio of the number of outcomes favorable to \(E\) to the total number of possible outcomes in the sample space, provided all outcomes are equally likely. Mathematically, this is expressed as: \[ P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Step 1: Identify the total size of the sample space.
From the problem statement, we are looking at a specific colony where a survey of smokers was conducted. The total number of people identified as smokers in this colony is given as: \[ \text{Total number of smokers (Total outcomes, } n(S)\text{)} = 50 \] The random selection is made strictly from this group of \(50\) smokers.

Step 2: Determine the number of favorable outcomes (number of female smokers).
The problem states that out of these \(50\) smokers, the number of male smokers is explicitly \(30\): \[ \text{Number of male smokers} = 30 \] Since the group of smokers consists of only males and females, we can calculate the number of female smokers by subtracting the number of male smokers from the total number of smokers: \[ \text{Number of female smokers (Favorable outcomes, } n(E)\text{)} = \text{Total smokers} - \text{Male smokers} \] \[ n(E) = 50 - 30 = 20 \]

Step 3: Calculate the probability.
Let \(E\) be the event that the randomly selected smoker is a female. Applying the classical definition of probability: \[ P(E) = \frac{\text{Number of female smokers}}{\text{Total number of smokers}} \] \[ P(E) = \frac{20}{50} \] To simplify the fraction, we divide both the numerator and the denominator by their greatest common divisor, which is \(10\): \[ P(E) = \frac{20 \div 10}{50 \div 10} = \frac{2}{5} \] Thus, the probability that the selected person is a female is \(\frac{2}{5}\), which matches option (B).

Answer 2

Concept: The complement rule of probability states that the probability of an event not occurring is equal to \(1\) minus the probability of that event occurring. If \(P(E)\) represents the probability that an event occurs, then the probability of the complementary event \(P(E')\) or \(P(\text{not } E)\) is given by: \[ P(E') = 1 - P(E) \] In this question, we are dealing with conditional probability. We are given that a male person has already been chosen, so our sample space is restricted exclusively to the population of smoking males.

Step 1: Identify the given probability for a male developing lung complications.
From the problem statement, the clinical study data provides the rate of lung complications among the smoking male population: \[ 170 \text{ out of } 1000 \text{ males who smoke develop lung complications.} \] Let \(L\) be the event that a person develops lung complications, and let \(M\) be the event that the person is male. The conditional probability that a selected male person suffers from lung problems is written as \(P(L|M)\): \[ P(L|M) = \frac{170}{1000} \] Dividing the numerator and the denominator by \(10\), we simplify the fraction to: \[ P(L|M) = \frac{17}{100} \]

Step 2: Apply the complement rule to find the probability of not suffering from lung complications.
We want to find the probability that a selected male person does not suffer from lung problems, which is denoted as \(P(L'|M)\). According to the complement rule: \[ P(L'|M) = 1 - P(L|M) \] Substituting the simplified value of \(P(L|M) = \frac{17}{100}\) into the equation: \[ P(L'|M) = 1 - \frac{17}{100} \] To perform the subtraction, convert the whole number \(1\) into a fraction with a common denominator of \(100\): \[ P(L'|M) = \frac{100}{100} - \frac{17}{100} \] Subtract the numerators over the common denominator: \[ P(L'|M) = \frac{100 - 17}{100} = \frac{83}{100} \] Thus, the probability that a chosen male person will not suffer from lung problems is \(\frac{83}{100}\), which matches option (B).

Answer 3

Concept: This problem requires the application of Bayes' Theorem. Bayes' Theorem allows us to find the conditional probability of an event (in this case, being a female, \(F\)) given that another event has already occurred (the person has lung complications, \(L\)). The formula for Bayes' Theorem is: \[ P(F|L) = \frac{P(F) \cdot P(L|F)}{P(M) \cdot P(L|M) + P(F) \cdot P(L|F)} \] where:
• \(P(M)\) and \(P(F)\) are the prior probabilities of selecting a male and a female smoker, respectively.
• \(P(L|M)\) and \(P(L|F)\) are the conditional probabilities of developing lung complications given that the person is a male or a female.
• The denominator represents the total probability of a person having lung complications, \(P(L)\).

Step 1: Identify all prior and conditional probabilities from the given data.
From the previous sub-parts, we have the following data for the colony of \(50\) smokers (\(30\) males, \(20\) females):
• Probability of selecting a male smoker: \(P(M) = \frac{30}{50} = \frac{3}{5}\)
• Probability of selecting a female smoker: \(P(F) = \frac{20}{50} = \frac{2}{5}\) From the general study, the conditional probabilities of suffering from lung complications are:
• For males: \(P(L|M) = \frac{170}{1000} = \frac{17}{100}\)
• For females: \(P(L|F) = \frac{120}{1000} = \frac{12}{100}\)

Step 2: Compute the total probability of a person having lung complications, \(P(L)\).
The denominator of Bayes' Theorem is the total probability \(P(L)\), which can happen via two mutually exclusive paths (either the person is a male with lung complications or a female with lung complications): \[ P(L) = P(M) \cdot P(L|M) + P(F) \cdot P(L|F) \] Substitute the values into the equation: \[ P(L) = \left( \frac{3}{5} \times \frac{17}{100} \right) + \left( \frac{2}{5} \times \frac{12}{100} \right) \] Calculate each product carefully: \[ \text{First term (Male path)} = \frac{3 \times 17}{5 \times 100} = \frac{51}{500} \] \[ \text{Second term (Female path)} = \frac{2 \times 12}{5 \times 100} = \frac{24}{500} \] Add the fractions together since they share a common denominator: \[ P(L) = \frac{51 + 24}{500} = \frac{75}{500} \]

Step 3: Apply Bayes' Theorem to find \(P(F|L)\).
Now, substitute the female path product (numerator) and total probability (denominator) into Bayes' formula: \[ P(F|L) = \frac{P(F) \cdot P(L|F)}{P(L)} = \frac{\frac{24}{500}}{\frac{75}{500}} \] We can cancel out the common denominator of \(500\) from both the numerator and the denominator: \[ P(F|L) = \frac{24}{75} \] To simplify this fraction further, find the greatest common divisor of \(24\) and \(75\), which is \(3\): \[ P(F|L) = \frac{24 \div 3}{75 \div 3} = \frac{8}{25} \] Hence, the conditional probability that the person is a female given they have lung problems is \(\frac{8}{25}\), matching option (A).

Answer 4

Concept: This problem is another variation of Bayes' Theorem, where we calculate the inverse probability that a randomly chosen person is male (\(M\)) given that the observed condition is that they do not have lung complications (\(L'\)). The formula for this conditional probability is: \[ P(M|L') = \frac{P(M) \cdot P(L'|M)}{P(M) \cdot P(L'|M) + P(F) \cdot P(L'|F)} \]

Step 1: Establish the complement conditional probabilities.
From the previous parts, we know the probabilities of being a male or female smoker in the colony: \[ P(M) = \frac{3}{5}, \quad P(F) = \frac{2}{5} \] We need the conditional probabilities of not having lung complications. Using the complement rule:
• For males: \(P(L'|M) = 1 - P(L|M) = 1 - \frac{170}{1000} = \frac{830}{1000} = \frac{83}{100}\)
• For females: \(P(L'|F) = 1 - P(L|F) = 1 - \frac{120}{1000} = \frac{880}{1000} = \frac{88}{100}\)

Step 2: Compute the total probability of a person not having lung complications, \(P(L')\).
We can compute the denominator \(P(L')\) either by summing the healthy paths or by taking the complement of the total sick probability \(P(L)\) found in part (a). Let's use both to show a highly descriptive breakdown. Using the total probability formula: \[ P(L') = P(M) \cdot P(L'|M) + P(F) \cdot P(L'|F) \] Substitute the fractions into the expression: \[ P(L') = \left( \frac{3}{5} \times \frac{83}{100} \right) + \left( \frac{2}{5} \times \frac{88}{100} \right) \] Calculate each product explicitly: \[ \text{Male healthy path} = \frac{3 \times 83}{500} = \frac{249}{500} \] \[ \text{Female healthy path} = \frac{2 \times 88}{500} = \frac{176}{500} \] Summing these fractions: \[ P(L') = \frac{249 + 176}{500} = \frac{425}{500} \] *(Alternative check: From part (a), \(P(L) = \frac{75}{500}\). Therefore, \(P(L') = 1 - \frac{75}{500} = \frac{425}{500}\). Both paths match perfectly!)*

Step 3: Apply Bayes' Theorem to solve for \(P(M|L')\).
Substitute the male healthy path (numerator) and total healthy probability (denominator) into the conditional formula: \[ P(M|L') = \frac{\frac{249}{500}}{\frac{425}{500}} \] Canceling out the common denominator of \(500\) yields: \[ P(M|L') = \frac{249}{425} \] Let us verify if this fraction can be further simplified. The prime factors of \(425\) are \(5\) and \(17\) (\(425 = 25 \times 17\)). Testing divisibility for \(249\): it is not divisible by \(5\), nor is it divisible by \(17\) (\(249 \div 17 \approx 14.64\)). Thus, the fraction is in its simplest irreducible form. This matches option (A) perfectly.