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Q3. An insurance company insures burglary insurance of families from 4 districts...

Apr 9, 2024

Q3. An insurance company insures burglary insurance of families from 4 districts. Here below are the statistics on the insured families of the company: \begin{tabular}{|c|c|c|} \hline District & Probability of burglary & Portion of the insured families \\ \hline A & 0.05 & 0.20 \\ B & 0.02 & 0.10 \\ C & 0.03 & 0.48 \\ D & 0.01 & 0.25 \\ \hline \end{tabular} Table 1: Statistics of the insured families There is now a randomly selected family insured by the company that has suffered a burglary accident. What is the probability that the insured family is from District A? (5 marks)

Solution by Steps

step 1

To find the probability that the insured family is from District A given that a burglary has occurred, we use Bayes' theorem

step 2

Bayes' theorem states that

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}

step 3

Here,

P(A)

is the portion of insured families from District A,

P(B|A)

is the probability of burglary in District A, and

P(B)

is the total probability of a burglary

step 4

From the table,

P(A) = 0.20

and

P(B|A) = 0.05

step 5

To calculate

P(B)

, we sum the probabilities of a burglary in each district weighted by the portion of insured families in that district:

P(B) = P(B|A) \cdot P(A) + P(B|B) \cdot P(B) + P(B|C) \cdot P(C) + P(B|D) \cdot P(D)

step 6

Substituting the values from the table into the equation for

P(B)

, we get

P(B) = 0.05 \cdot 0.20 + 0.02 \cdot 0.10 + 0.03 \cdot 0.48 + 0.01 \cdot 0.25

step 7

Calculating

P(B)

, we find

P(B) = 0.01 + 0.002 + 0.0144 + 0.0025 = 0.0289

step 8

Now we can apply Bayes' theorem to find

P(A|B)

P(A|B) = \frac{0.05 \cdot 0.20}{0.0289}

step 9

Calculating

P(A|B)

, we get

P(A|B) = \frac{0.01}{0.0289} \approx 0.346

Answer

The probability that the insured family is from District A given that a burglary has occurred is approximately 0.346.

Key Concept

Bayes' theorem

Explanation

Bayes' theorem is used to find the conditional probability of an event given that another event has occurred. It relates the conditional and marginal probabilities of stochastic events.