Explanation

Bayes' formula states:

$P(X|I) = \frac{P(I|X) \times P(X)}{P(I)}$

Where:

• $P(X|I)$ = Updated probability of X given the new information
• $P(I|X)$ = Probability of the new information given X = 0.50
• $P(X)$ = Prior probability of X = 0.70
• $P(I)$ = Unconditional probability of the new information = 0.60

Plugging in the values:

$P(X|I) = \frac{0.50 \times 0.70}{0.60} = \frac{0.35}{0.60} = 0.5833$

Rounded to two decimal places, this is approximately 0.58.

Step-by-step calculation:

1. Multiply $P(I|X)$ and $P(X)$: $0.50 \times 0.70 = 0.35$
2. Divide by $P(I)$: $0.35 \div 0.60 = 0.5833$
3. The result is closest to 0.58 (Option B)

Why not the other options?

• Option A (0.43): This would be $\frac{0.50 \times 0.70}{0.80}$ or similar incorrect calculation
• Option C (0.84): This would be $\frac{0.60 \times 0.70}{0.50}$ or similar incorrect calculation

Key Concept: Bayes' theorem allows us to update prior probabilities with new information to obtain posterior probabilities, which is fundamental in probability theory and statistical inference.