**The probabilities of the independent events A and B are 0.4 and 0.5 respectively. Find the following probabilities:**

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a) The probability of A and B occurring.

b) The probability of A but not B occurring.

c) The probability of A or B occurring.

d) The probability of A or not B occurring.

e) The probability of A occurring given that B has occurred.

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**f) The probability of A occurring given that B has not occurred.**