Probability with replacement vs without replacement

A question is this type if and only if it explicitly contrasts or involves scenarios where items are either replaced or not replaced between selections.

2 questions · Standard +0.3

2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles
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CAIE S1 2024 March Q1
4 marks Standard +0.3
1 A bag contains 9 blue marbles and 3 red marbles. One marble is chosen at random from the bag. If this marble is blue, it is replaced back into the bag. If this marble is red, it is not returned to the bag. A second marble is now chosen at random from the bag.
  1. Find the probability that both the marbles chosen are red.
  2. Find the probability that the first marble chosen is blue given that the second marble chosen is red.
CAIE S1 2024 November Q4
6 marks Standard +0.3
4 Rahul has two bags, \(X\) and \(Y\). Bag \(X\) contains 4 red marbles and 2 blue marbles. Bag \(Y\) contains 3 red marbles and 4 blue marbles. Rahul also has a coin which is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). Rahul throws the coin.
  • If he obtains a head, he chooses at random a marble from bag \(X\). He notes the colour and replaces the marble in bag \(X\). He then chooses at random a second marble from bag \(X\).
  • If he obtains a tail, he chooses at random a marble from bag \(Y\). He notes the colour and discards the marble. He then chooses at random a second marble from bag \(Y\).
    1. Find the probability that the two marbles that Rahul chooses are the same colour. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-06_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-07_2725_35_99_20}
    2. Find the probability that the two marbles that Rahul chooses are both from bag \(Y\) given that both marbles are blue.