Answered

This year the CDC reported that 30% of adults received
their flu shot. Of those adults who received their flu shot,
20% still got the flu. Of those who did not receive the flu
shot, 65% got the flu.
What's the probability that a randomly selected adult
got the flu?
Enter your answer as a decimal to the ten thousandths.
place.
0.20
Flu
Flu shot
0.3
0.80
No flu
Adult
0.65
Flu
0.7
No flu shot
0.35
No flu

Verified Answer :

GinnyAnswer

To determine the probability that a randomly selected adult got the flu, we apply the concept of total probability with the provided data. Let's define the relevant probabilities and compute the required probability step by step.

### Given Data:
1. Probability that an adult received the flu shot (\(\(P(\text{Flu shot})\)\)): \(\(0.3\)\)
2. Probability that an adult did not receive the flu shot (\(\(P(\text{No flu shot})\)\)): \(\(1 - 0.3 = 0.7\)\)
3. Probability of getting the flu given that the adult received the flu shot (\(\(P(\text{Flu} \mid \text{Flu shot})\)\)): \(\(0.2\)\)
4. Probability of getting the flu given that the adult did not receive the flu shot (\(\(P(\text{Flu} \mid \text{No flu shot})\)\)): \(\(0.65\)\)

### Applying Total Probability:

The total probability of an adult getting the flu can be found using the Law of Total Probability:
\(\[ P(\text{Flu}) = P(\text{Flu shot}) \cdot P(\text{Flu} \mid \text{Flu shot}) + P(\text{No flu shot}) \cdot P(\text{Flu} \mid \text{No flu shot}) \]\)

### Calculations:

1. Calculate the probability that an adult got the flu and received the flu shot:
\(\[ P(\text{Flu shot}) \cdot P(\text{Flu} \mid \text{Flu shot}) = 0.3 \cdot 0.2 = 0.06 \]\)

2. Calculate the probability that an adult got the flu and did not receive the flu shot:
\(\[ P(\text{No flu shot}) \cdot P(\text{Flu} \mid \text{No flu shot}) = 0.7 \cdot 0.65 = 0.455 \]\)

3. Add these two probabilities to get the total probability that an adult got the flu:
\(\[ P(\text{Flu}) = 0.06 + 0.455 = 0.515 \]\)

Thus, the probability that a randomly selected adult got the flu is \(\(0.515\)\).

### Answer:
The probability that a randomly selected adult got the flu is \(\(0.5150\)\).

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