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. $4 Points$

Gerhard
4.5
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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