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



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

### Applying Total Probability:

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

### Calculations:

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

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

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

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

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

Other Questions