Question

A pandemic has been spreading all over the world. The probabilities are 0.7 that there will be a lockdown , 0.8 that the pandemic is controlled in one month if there is a lockdown and 0.3 that it is controlled in one month if there is no lockdown. the probability that the pandemic will be controlled in one month is

• A. 0.65
• B. 1.46
• C. 1.65
• D. 0.46

Answer 1

The Correct Option is A

Let: 
\( P(L) = 0.7 \) → Probability of lockdown 
\( P(\overline{L}) = 1 - 0.7 = 0.3 \) → No lockdown 
\( P(C|L) = 0.8 \) → Probability that pandemic is controlled in one month given lockdown 
\( P(C|\overline{L}) = 0.3 \) → Probability that pandemic is controlled in one month given no lockdown 

Using the law of total probability: 
\[ P(C) = P(L) \cdot P(C|L) + P(\overline{L}) \cdot P(C|\overline{L}) \] \[ P(C) = (0.7)(0.8) + (0.3)(0.3) = 0.56 + 0.09 = 0.65 \] Correct Answer: 0.65

Answer 2

Let L be the event that there will be a lockdown. 

Let C be the event that the pandemic is controlled in one month.

We are given the following probabilities:

• The probability of a lockdown: \(P(L) = 0.7\)
• The probability of no lockdown: \(P(L') = 1 - P(L) = 1 - 0.7 = \mathbf{0.3}\)
• The probability that the pandemic is controlled given there is a lockdown: \(P(C|L) = 0.8\)
• The probability that the pandemic is controlled given there is no lockdown: \(P(C|L') = 0.3\)

We want to find the probability that the pandemic will be controlled in one month, which is \(P(C)\).

We can use the Law of Total Probability. The event C can occur in two mutually exclusive ways: either there is a lockdown and the pandemic is controlled (\(C \cap L\)), or there is no lockdown and the pandemic is controlled (\(C \cap L'\)).

\[ P(C) = P(C \cap L) + P(C \cap L') \]

Using the definition of conditional probability, \(P(C \cap L) = P(C|L)P(L)\) and \(P(C \cap L') = P(C|L')P(L')\).

So, the formula becomes:

\[ \mathbf{P(C) = P(C|L)P(L) + P(C|L')P(L')} \]

Substitute the given values into the formula:

\[ P(C) = (0.8)(0.7) + (0.3)(0.3) \]

\[ P(C) = 0.56 + 0.09 \]

\[ P(C) = \mathbf{0.65} \]

The probability that the pandemic will be controlled in one month is 0.65.

Comparing this with the given options, the correct option is:

0.65