First success before/after trial n

7 A children's wildlife magazine is published every Monday. For the next 12 weeks it will include a model animal as a free gift. There are five different models: tiger, leopard, rhinoceros, elephant and buffalo, each with the same probability of being included in the magazine. Sahim buys one copy of the magazine every Monday.

1. Find the probability that the first time that the free gift is an elephant is before the 6th Monday.
2. Find the probability that Sahim will get more than two leopards in the 12 magazines.
3. Find the probability that after 5 weeks Sahim has exactly one of each animal.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 Three fair 6-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time repeatedly. The score on each throw is the sum of the numbers on the uppermost faces.

1. Find the probability that a score of 17 or more is first obtained on the 6th throw.
2. Find the probability that a score of 17 or more is obtained in fewer than 8 throws.

5 A factory produces chocolates. 30\% of the chocolates are wrapped in gold foil, 25\% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.

1. Find the probability that she obtains no more than 2 chocolates that are wrapped in red foil.
   Jake chooses chocolates one at a time at random from the production line.
2. Find the probability that the first time he obtains a chocolate that is wrapped in red foil is before the 7th choice. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-08_2720_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-09_2717_29_105_22} Keifa chooses chocolates one at a time at random from the production line.
3. Find the probability that the second chocolate chosen is the first one wrapped in gold foil given that the fifth chocolate chosen is the first unwrapped chocolate.

9

1. A clock is designed to chime once each hour, on the hour. The clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 10 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day. Find the probability that the first time it does not chime is
   1. at 0600 on that day,
   2. before 0600 on that day.
   3. Another clock is designed to chime twice each hour: on the hour and at 30 minutes past the hour. This clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 20 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day.
      (a) Find the probability that the first time it does not chime is at either 0030 or 0130 on that day.
      (b) Use the formula for the sum to infinity of a geometric progression to find the probability that the first time it does not chime is at 30 minutes past some hour.