An insurer records the cause of each crop-damage claim in a region as flood, drought or frost. The table shows the probability of each cause. Cause Flood Dr...
Question 1 Report
An insurer records the cause of each crop-damage claim in a region as flood, drought or frost. The table shows the probability of each cause.
| Cause | Flood | Drought | Frost |
|---|---|---|---|
| Probability | 0.28 | \(2x\) | \(x\) |
- Find the value of \(x\). (1)
- Find the probability that a random claim is due to drought. (1)
- The insurer received 125 claims this year. Work out how many were due to frost. (2)
Answer Details
This question tests setting up and solving an equation from the fact that all outcome probabilities sum to 1, then using the resulting probabilities for a simple probability and an expected frequency.
(a) Flood, drought and frost are the only causes, so their probabilities sum to 1:
\[ 0.28 + 2x + x = 1 \]
\[ 3x = 1 - 0.28 = 0.72 \]
\[ x = \frac{0.72}{3} = 0.24 \] [1 mark]
(b) The probability of drought is \( 2x \), so substitute the value of \( x \) found in part (a):
\[ P(\text{drought}) = 2 \times 0.24 = 0.48 \] [1 mark]
(c) The probability of frost is \( x = 0.24 \); multiply this by the total number of claims to estimate how many were due to frost:
\[ \text{Frost claims} = 125 \times 0.24 = 30 \] [2 marks]
As a check, all three probabilities sum correctly: \( 0.28 + 0.48 + 0.24 = 1 \).
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