Independence test with P(A∩B) = P(A)×P(B)

2. A car dealer offers purchasers a three year warranty on a new car. He sells two models, the Zippy and the Nifty. For the first 50 cars sold of each model the number of claims under the warranty is shown in the table below. One of the purchasers is chosen at random. Let \(A\) be the event that no claim is made by the purchaser under the warranty and \(B\) the event that the car purchased is a Nifty.

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A ^ { \prime } \right)\). Given that the purchaser chosen does not make a claim under the warranty,
3. find the probability that the car purchased is a Zippy.
4. Show that making a claim is not independent of the make of the car purchased. Comment on this result.

In a large examination room each candidate has just one electronic calculator.

• \(G\) is the event that a randomly chosen candidate has a graphical calculator.
• \(T\) is the event that a randomly chosen candidate has a 'Texio' brand calculator.

You are given the following probabilities. $$\text{P}(G) = 0.65 \quad \text{P}(T) = 0.4 \quad \text{P}(G \cap T) = 0.25$$

1. Are the events \(G\) and \(T\) independent? Justify your answer with an appropriate calculation. [2]
2. Find P(\(T | G\)) and explain, in the context of this question, what this probability represents. [3]