2025 RI P2 Q10

A bag contains \(4\) identical red balls and \(6\) identical blue balls.

1. In the first game, the balls are randomly picked by a player from the bag, one at a time, without replacement, until there are no balls left in the bag.
   1. Find the probability that the last ball picked is red.[1]
   2. Find the probability that in the first five picks, exactly \(2\) blue balls are picked given that at least \(3\) red balls are picked.[3]
2. In the second game, each ball picked by a player will be placed back into the bag before the next ball is picked. The player makes a total of \(20\) random picks from the bag and the colour of each pick is recorded.
   1. Find the probability that the colour red is recorded exactly \(4\) times.[2]
   2. Find the probability that the colour red is recorded more than \(4\) times but not more than \(8\) times.[2]

   3. Find the probability that the \(8 ^\mathrm{th}\) pick is the \(6^\mathrm{th}\) time the colour blue is recorded.

      [2]
   4. The second game is played by \(50\) randomly chosen players. Estimate the probability that the average number of times the colour red is recorded is more than \(8.5\).[4]