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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Gerhard

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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%.

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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