Question

(a) The probability that an automobile being filled with gasoline will also need an oil change is 0.35; the probability that it needs a new oil filter is 0.45, and the probability that both the oil and filter need changing is 0.25. (i) If the oil had to be changed, what is the probability that a new oil filter is needed? (ii) If a new oil filter is needed, what is the probability that the oil has to be changed? (b) 3 Cards are drawn in succession without replacement from an ordinary deck. Find the probability of the event A1 ∩ A2 ∩ A3, where A1 is the event that the first card is RED ace, A2 is the event that the second card Queen or King, and A3 is the event that the third card which is greater than 3 and less than Eight?

Answer 1

0.714 ; 0.556 ; (A1 n A2 n A3) = 0

Explanation:

Given that :

P(oil) = 0.35

P(filter) = 0.45

P(oil n filter) = 0.25

1)

P(filter | Oil) = P(oil n filter) / P(oil)

P(filter | Oil) = 0.25 / 0.35 = 0.714

11)

P(Oil | filter) = P(oil n filter) / P(filter)

P(Oil | filter ) = 0.25 / 0.45 = 0.556

Number of red aces = 2

Queen or king = 4 + 4 = 8

Number on card greater than 3 and less than 8 = {4,5,6,7} = 4 card ; for 4 suits = 4 * 4 = 16

Total number of cards in deck = 52

Choosing without replacement :

Hence,

A1 = 2 / 52

A2 = 8 / 51

A3 = 16 / 50

(A1 n A2 n A3) ;this means card common to all three events.

However, (A1 n A2 n A3) = 0 because ;

1.) No red ace card is greater than 3 and less than 8.

11.) An ace card can neither be a king nor a queen