- The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)
When more than one game is played the games are independent.
Sam plays 20 games.
- Find the probability that Sam wins 4 or more prizes.
Tessa plays a series of games.
- Find the probability that Tessa wins her 4th prize on her 20th game.
Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) )
Sam decides to play \(n\) games of Pento with Rama.
- Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\)
Given that Sam chose \(n = 15\)
- find the probability that Sam does not make a loss.
Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
- Find, in terms of \(r\), Tessa's expected profit.