Geometric Probability

1. Two people are playing darts. Peg hits points randomly on the circular board, whose radius is \(a\). If the distance from the centre \(O\) of the point that she hits is modelled by the variable \(R\),
   1. explain why the cumulative distribution function \(\mathrm { F } ( r )\) is given by
   $$\begin{array} { l l } \mathrm { F } ( r ) = 0 & r < 0 ,
   \mathrm {~F} ( r ) = \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leq r \leq a ,
   \mathrm {~F} ( r ) = 1 & r > a . \end{array}$$
2. By first finding the probability density function of \(R\), show that the mean distance from \(O\) of the points that Peg hits is \(\frac { 2 a } { 3 }\). Bob, a more experienced player, aims for \(O\), and his points have a distance \(X\) from \(O\) whose cumulative distribution function is $$\mathrm { F } ( x ) = 0 , x < 0 ; \quad \mathrm { F } ( x ) = \frac { x } { a } \left( 2 - \frac { x } { a } \right) , 0 \leq x \leq a ; \quad \mathrm { F } ( x ) = 1 , x > a .$$
3. Find the probability density function of \(X\), and explain why it shows that Bob is aiming for \(O\).

1. A string of length 40 cm is cut into 2 pieces at a random point. The continuous random variable \(L\) represents the length of the longer piece of string.
   1. Write down the distribution of \(L\)
   2. Find the probability that the length of the longer piece of string is 28 cm to the nearest cm
   Each piece of string is used to form the perimeter of a square.
2. Calculate the probability that the area of the larger square is less than \(64 \mathrm {~cm} ^ { 2 }\)
3. Calculate the probability that the difference in area between the two squares is greater than \(81 \mathrm {~cm} ^ { 2 }\)

1 Abby runs a stall at a charity event. Visitors to the stall pay to play a game in which six fair dice are rolled. If the difference between the highest and lowest scores is less than 3 then the player wins \(\pounds 5\). Otherwise the player wins nothing. Abby designs the spreadsheet shown in Fig. 1 to estimate the probability of a player winning, by simulating 20 goes at the game. Cell C5, highlighted, shows that the 2nd dice in simulated game 4 scores 5 . Cells H5 and I5 show the highest and lowest scores, respectively, in game 4, and cell J5 gives the difference between them. \begin{table}[h] \end{table}

1. (A) Write down the numbers in columns H , I and J for game 20 .
   (B) Use the spreadsheet to estimate the probability of a player winning a game.
2. State how the estimate of probability in (i) (B) could be improved.
3. Give one advantage and one disadvantage of using this simulation technique compared with working out the theoretical probability. All profit made by the stall is given to charity. Abby has to decide how much to charge players to play.
4. If Abby charges \(\pounds 1\) per game, estimate the total profit when 50 players each play the game once.