Probability with replacement vs without replacement

4 The probability that it will rain on any given day is \(x\). If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3 . Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4 . If he does not wear a hat, the probability that he wears a scarf is 0.1 . The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36 . Find the value of \(x\).

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.

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.
     

13 There are two types of coins in a money box:

• 20\% are bronze coins
• 80\% are silver coins

Craig takes out a coin at random and places it back in the money box.
Craig then takes out a second coin at random.
13

1. Find the probability that both coins were of the same type.
   13
2. Find the probability that both coins are bronze, given that at least one of the coins is bronze.
   \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-23_2488_1716_219_153} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}