Question

The person taking vaccine \( A_2 \) will get infected.

Answer 1

Concept:
• Probability of an event and its complement: \( P(E') = 1 - P(E) \).
• Given a conditional probability of success (protection), the probability of failure (infection) is its complement.

Step 1: Identify the given probabilities for Vaccine \( A_2 \)
From the case study, for a person taking Vaccine \( A_2 \): Probability of being protected, \( P(\text{Protected} | A_2) = 55\% = 0.55 \).

Step 2: Calculate the probability of being infected
Getting infected is the complementary event of being protected. \[ P(\text{Infected} | A_2) = 1 - P(\text{Protected} | A_2) \] \[ P(\text{Infected} | A_2) = 1 - 0.55 \] \[ P(\text{Infected} | A_2) = 0.45 \]

Step 3: Final Result
The probability that a person taking Vaccine \( A_2 \) will get infected is \( 0.45 \) or \( 45\% \).

Answer 2

Concept:
• Theorem of Total Probability: For mutually exclusive events \( E_1, E_2, E_3 \), the total probability of an event \( A \) is \( P(A) = \sum P(E_i)P(A|E_i) \).

Step 1: Define the events and their probabilities
Let \( E_1, E_2, E_3 \) be the events that a person takes Vaccine \( A_1, A_2, A_3 \) respectively. \[ P(E_1) = 25\% = 0.25 \] \[ P(E_2) = 35\% = 0.35 \] \[ P(E_3) = 40\% = 0.40 \] Let \( A \) be the event that the person is protected. \[ P(A|E_1) = 0.60 \] \[ P(A|E_2) = 0.55 \] \[ P(A|E_3) = 0.50 \]

Step 2: Apply the Total Probability Formula
\[ P(A) = P(E_1)P(A|E_1) + P(E_2)P(A|E_2) + P(E_3)P(A|E_3) \] \[ P(A) = (0.25 \times 0.60) + (0.35 \times 0.55) + (0.40 \times 0.50) \]

Step 3: Calculate the final numerical value
\[ P(A) = 0.1500 + 0.1925 + 0.2000 \] \[ P(A) = 0.5425 \] The probability that a randomly chosen person is protected is \( 0.5425 \).

Answer 3

Concept:
• Bayes' Theorem: \( P(E_k|I) = \frac{P(E_k)P(I|E_k)}{P(I)} \).
• \( P(I) = 1 - P(\text{Protected}) \).

Step 1: Calculate the total probability of being infected
From question 37(ii), total probability of protection \( P(A) = 0.5425 \). Total probability of infection \( P(I) = 1 - 0.5425 = 0.4575 \).

Step 2: Calculate the numerator for Bayes' Theorem
We need the probability of being infected given Vaccine \( A_1 \). \[ P(I|E_1) = 1 - 0.60 = 0.40 \] Numerator: \( P(E_1)P(I|E_1) = 0.25 \times 0.40 = 0.10 \).

Step 3: Apply Bayes' Theorem
\[ P(E_1|I) = \frac{P(E_1)P(I|E_1)}{P(I)} \] \[ P(E_1|I) = \frac{0.10}{0.4575} \] Converting to fraction: \[ P(E_1|I) = \frac{1000}{4575} = \frac{40}{183} \]

Answer 4

Concept:

• Bayes' Theorem for protected individuals (not infected): \( P(E_3|A) = \frac{P(E_3)P(A|E_3)}{P(A)} \).

Step 1: Identify the required values from previous steps
\( P(A) = 0.5425 \) (Total Probability of protection).
\( P(E_3) = 0.40 \).
\( P(A|E_3) = 0.50 \).

Step 2: Calculate the numerator
\[ P(E_3)P(A|E_3) = 0.40 \times 0.50 = 0.20 \]

Step 3: Apply Bayes' Theorem
\[ P(E_3|A) = \frac{0.20}{0.5425} \]
Converting to fraction:
\[ P(E_3|A) = \frac{2000}{5425} = \frac{80}{217} \]