John bought a car a number of years ago. The table below gives an estimate of the probability that each of the following three events happens to John's car in the next year. Event Probability Head gasket blows 0.095 Timing belt goes 0.041 Air filters break 0.073 (i) If the head gasket blows, John will have to replace his car, at an estimated cost of €20000. If the head gasket is replaced now, it will cost €1450, and the probability that it blows in the next year will be reduced to 0.005. Based on these figures, use expected values to work out if it is worth replacing the head gasket now, or not. (ii) Work out the probability that at least one of the events in the table above happens to John’s car this year, taking these events to be independent. Give your answer correct to 3 decimal places.

Leaving Cert Mathematics Probability: John bought a car a number of ye

John bought a car a number of years ago - Leaving Cert Mathematics - Question d - 2022

Question d

John bought a car a number of years ago. The table below gives an estimate of the probability that each of the following three events happens to John's car in the ne... show full transcript

Worked Solution & Example Answer:John bought a car a number of years ago - Leaving Cert Mathematics - Question d - 2022

Step 1

If the head gasket blows, John will have to replace his car.

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Answer

To determine whether to replace the head gasket now or not, we need to calculate the expected values for both scenarios.

When not replacing the head gasket:

The expected cost can be calculated using the formula: $E(X) = P(HG) imes ext{Cost of Replacement}$ where:
- $P(HG) = 0.095$
- Cost of Replacement = €20000.
Therefore, $E(X) = 0.095 imes 20000 = 1900.$
When replacing the head gasket:

The expected cost in this scenario will be: $E(X) = 1450 + P(HG_{new}) imes ext{Cost of Replacement}$ where:
- $P(HG_{new}) = 0.005$
- Cost of Replacement = €20000.
Thus, $E(X) = 1450 + (0.005 imes 20000) = 1450 + 100 = 1550.$

Conclusion: John should replace the head gasket now since the expected cost is lower at €1550 compared to €1900.

Step 2

Work out the probability that at least one of the events in the table above happens to John’s car this year.

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Answer

To find the probability that at least one event occurs, we can first determine the probability that none of the events happen. The events are independent, so:

Calculating the probability of none of the events occurring:

The probability of each event not occurring is:
- For head gasket blowing: $1 - 0.095 = 0.905$ .
- For timing belt going: $1 - 0.041 = 0.959$ .
- For air filters breaking: $1 - 0.073 = 0.927$ .
Therefore, the total probability of none occurring is: $P( ext{none}) = 0.905 imes 0.959 imes 0.927.$
Calculating this value:
$= 0.8045 \ = 0.19547.$$$

So, the probability that at least one of the events occurs is: $P( ext{at least 1}) = 1 - P( ext{none}) = 1 - 0.8045 = 0.1954.$

Therefore, the answer is: $0.195$ (to 3 decimal places).

John bought a car a number of years ago - Leaving Cert Mathematics - Question d - 2022

Worked Solution & Example Answer:John bought a car a number of years ago - Leaving Cert Mathematics - Question d - 2022

If the head gasket blows, John will have to replace his car.

Work out the probability that at least one of the events in the table above happens to John’s car this year.

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