Question

A vaccine has a 90% probability of being effective in preventing a certain disease. The probability of getting the disease if a person is not vaccinated is 50%. In a certain geographic region, 60% of the people get vaccinated. If a person is selected at random from this region, find the probability that he or she will contract the disease. (4 Points)

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Answer to a math question A vaccine has a 90% probability of being effective in preventing a certain disease. The probability of getting the disease if a person is not vaccinated is 50%. In a certain geographic region, 60% of the people get vaccinated. If a person is selected at random from this region, find the probability that he or she will contract the disease. (4 Points)

Expert avatar
Gerhard
4.5
43 Answers
Let's break down the problem into smaller parts.

Given:
- Probability of vaccine effectiveness (E) = 90%
- Probability of getting the disease without vaccination (D') = 50%
- Probability of getting vaccinated (V) = 60%

We need to find the probability that a person will contract the disease (D) when selected randomly from this region.

To find the probability of contracting the disease, we can use the law of total probability.

Step 1: Using the law of total probability, we can write the equation:

P(D) = P(D|E)V + P(D|E')V'

where:
- P(D) is the probability of getting the disease
- P(D|E) is the probability of getting the disease given that the vaccine is effective
- P(D|E') is the probability of getting the disease given that the vaccine is not effective
- V is the probability of getting vaccinated
- V' is the probability of not getting vaccinated, which is equal to 1 - V

Step 2: Substitute the given values into the equation:

P(D) = 0.9 * 0.6 + 0.5 * (1 - 0.6)

Step 3: Simplify the equation:

P(D) = 0.54 + 0.5 * 0.4

Step 4: Simplify further:

P(D) = 0.54 + 0.2

Step 5: Calculate:

P(D) = 0.74

Answer: The probability that a person will contract the disease is 0.74 or 74%.

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