Second success on trial n

5 Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.
[0pt]

1. Find the probability that Salah completes the puzzle for the first time on the 5th day. [1]
2. Find the probability that Salah completes the puzzle for the second time on the 5th day.
3. Find the probability that Salah completes the puzzle fewer than 5 times in a week (7 days). [3] \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-10_2713_31_145_2014}
4. Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.

1 Nicola throws an ordinary fair six-sided dice. The random variable \(X\) is the number of throws that she takes to obtain a 6.

1. Find \(\mathrm { P } ( X < 8 )\).
2. Find the probability that Nicola obtains a 6 for the second time on her 8th throw. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-02_2717_35_109_2012}

3 A child is trying to draw court cards from an ordinary pack of 52 cards (court cards are Kings, Queens and Jacks; there are 12 in a pack). She draws cards, one at a time, with replacement, from the pack. Find the probabilities of the following events.

1. She draws a court card for the first time on the sixth try.
2. She draws a court card at least once in the first six tries.
3. She draws a court card for the second time on the sixth try.
4. She draws at least two court cards in the first six tries.

1. The probability of Richard winning a prize in a game at the fair is 0.15

Richard plays a number of games.

1. Find the probability of Richard winning his second prize on his 8th game,
2. State two assumptions that have to be made, for the model used in part (a) to be valid. M ary plays the same game, but has a different probability of winning a prize. She plays until she has won r prizes. The random variable \(G\) represents the total number of games M ary plays.
3. Given that the mean and standard deviation of G are 18 and 6 respectively, determine whether Richard or Mary has the greater probability of winning a prize in a game.

The proportion of people who watch West Street on television is 30\%. A market researcher interviews people at random in order to contact viewers of West Street. Each day she has to contact a certain number of viewers of West Street.

1. Near the end of one day she finds that she needs to contact just one more viewer of West Street. Find the probability that the number of further interviews required is
   1. 4, [3]
   2. less than 4. [3]
2. Near the end of another day she finds that she needs to contact just two more viewers of West Street. Find the probability that the number of further interviews required is
   1. 5, [4]
   2. more than 5. [2]