On a Saturday night, 20%of all drivers in U.S.A. are under the influence of alcohol. The probability that a drive under the influence of alcohol will have an accident is 0.001. The probability that a sober drive will have an accident is 0.00. if a car on a Saturday night smashed into a tree, the probability that the driver was under the influence of alcohol is `3//7` b. `4//7` c. `5//7` d. `6//7`

Options

A. `3//7`
B. `4//7`
C. `5//7`
D. `6//7`

Explanation

To solve the problem, we will use Bayes' theorem. Let's break down the solution step by step.

### Step 1: Define the events
Let:
- \( A \) = event that a car had an accident.
- \( B_1 \) = event that the driver was under the influence of alcohol.
- \( B_2 \) = event that the driver was sober.

### Step 2: Given probabilities
From the problem, we know:
- \( P(B_1) = 0.2 \) (20% of drivers are under the influence of alcohol)
- \( P(B_2) = 1 - P(B_1) = 0.8 \) (80% of drivers are sober)
- \( P(A|B_1) = 0.001 \) (probability of an accident given the driver is under the influence)
- \( P(A|B_2) = 0.0001 \) (probability of an accident given the driver is sober)

### Step 3: Apply Bayes' theorem
We want to find \( P(B_1|A) \), the probability that the driver was under the influence of alcohol given that there was an accident. According to Bayes' theorem:

\[ P(B_1|A) = \frac{P(A|B_1) \cdot P(B_1)}{P(A)} \]

### Step 4: Calculate \( P(A) \)
To find \( P(A) \), we can use the law of total probability:

\[ P(A) = P(A|B_1) \cdot P(B_1) + P(A|B_2) \cdot P(B_2) \]

Substituting the known values:

\[ P(A) = (0.001 \cdot 0.2) + (0.0001 \cdot 0.8) \]

Calculating each term:

\[ P(A) = 0.0002 + 0.00008 = 0.00028 \]

### Step 5: Substitute back into Bayes' theorem
Now we can substitute \( P(A) \) back into the equation for \( P(B_1|A) \):

\[ P(B_1|A) = \frac{0.001 \cdot 0.2}{0.00028} \]

Calculating the numerator:

\[ 0.001 \cdot 0.2 = 0.0002 \]

Now substituting:

\[ P(B_1|A) = \frac{0.0002}{0.00028} = \frac{20}{28} = \frac{5}{7} \]

### Final Answer
Thus, the probability that the driver was under the influence of alcohol given that there was an accident is \( \frac{5}{7} \).