- Tisam is playing a game.
She uses a ball, a cup and a spinner.
The random variable \(X\) represents the number the spinner lands on when it is spun. The probability distribution of \(X\) is given in the following table
| \(x\) | 20 | 50 | 80 | 100 |
| \(\mathrm { P } ( X = x )\) | \(a\) | \(b\) | \(c\) | \(d\) |
where \(a , b , c\) and \(d\) are probabilities.
To play the game
- the spinner is spun to obtain a value of \(x\)
- Tisam then stands \(x \mathrm {~cm}\) from the cup and tries to throw the ball into the cup
The event \(S\) represents the event that Tisam successfully throws the ball into the cup.
To model this game Tisam assumes that
- \(\mathrm { P } ( S \mid \{ X = x \} ) = \frac { k } { x }\) where \(k\) is a constant
- \(\mathrm { P } ( S \cap \{ X = x \} )\) should be the same whatever value of \(x\) is obtained from the spinner
Using Tisam's model,
- show that \(c = \frac { 8 } { 5 } b\)
- find the probability distribution of \(X\)
Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm .
He successfully gets the ball in the cup \(30 \%\) of the time. - State, giving a reason, why Tisam's model of this game is not suitable to describe Nav playing the game for all values of \(X\)