Expected profit or cost problem

6 At a funfair, Amy pays \(
) 1$ for two attempts to make a bell ring by shooting at it with a water pistol.

• If she makes the bell ring on her first attempt, she receives \(
  ) 3\( and stops playing. This means that overall she has gained \)\\( 2\).
• If she makes the bell ring on her second attempt, she receives \(
  ) 1.50\( and stops playing. This means that overall she has gained \)\\( 0.50\).
• If she does not make the bell ring in the two attempts, she has lost her original \(
  ) 1$.

The probability that Amy makes the bell ring on any attempt is 0.2 , independently of other attempts.

1. Show that the probability that Amy loses her original \(
   ) 1$ is 0.64 .
2. Complete the probability distribution table for the amount that Amy gains.

   Amy's gain (\$)
   Probability	0.64

3. Calculate Amy's expected gain.

6 In a competition, people pay \(
) 1\( to throw a ball at a target. If they hit the target on the first throw they receive \)\\( 5\). If they hit it on the second or third throw they receive \(
) 3\(, and if they hit it on the fourth or fifth throw they receive \)\\( 1\). People stop throwing after the first hit, or after 5 throws if no hit is made. Mario has a constant probability of \(\frac { 1 } { 5 }\) of hitting the target on any throw, independently of the results of other throws.

1. Mario misses with his first and second throws and hits the target with his third throw. State how much profit he has made.
2. Show that the probability that Mario's profit is \(
   ) 0$ is 0.184 , correct to 3 significant figures.
3. Draw up a probability distribution table for Mario's profit.
4. Calculate his expected profit.

2 Two unbiased tetrahedral dice each have four faces numbered \(1,2,3\) and 4. The two dice are thrown together and the sum of the numbers on the faces on which they land is noted. Find the expected number of occasions on which this sum is 7 or more when the dice are thrown together 200 times.

6 A six-sided die is biased so that the probability of scoring 6 is 0.1 and the probabilities of scoring \(1,2,3,4\), and 5 are all equal. In a game at a fête, contestants pay \(\pounds 3\) to roll this die. If the score is 6 they receive \(\pounds 10\) back. If the score is 5 they receive \(\pounds 5\) back. Otherwise they receive no money back. Find the organiser's expected profit for 100 rolls of the die.

1 When a spinner is spun, the outcome is equally likely to be 1,2 or 3 . In a competition, the spinner is spun twice and the outcomes are added to give a total score \(T\).

1. Show that the expectation of \(T\) is 4 .
2. Find the variance of \(T\). A competitor pays \(\pounds 1.50\) to enter the competition and receives \(\pounds X\), where \(X = 0.3 T\).
3. 1. Find the expectation of the competitor's profit.
   2. Find the variance of the competitor's profit.

7 Each week, a newsagent stocks 5 copies of the magazine Statistics Weekly. A regular customer always buys one copy. The demand for additional copies may be modelled by a Poisson distribution with mean 2. The number of copies sold in a week, \(X\), has the probability distribution shown in the table, where probabilities are stated correct to three decimal places.

1. Show that, correct to three decimal places, the values of \(a\) and \(b\) are 0.180 and 0.143 respectively.
2. Find the values of \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\), showing the calculations needed to obtain these values, and hence calculate the standard deviation of \(X\).
3. The newsagent makes a profit of \(\pounds 1\) on each copy of Statistics Weekly that is sold and loses 50 p on each copy that is not sold. Find the mean weekly profit for the newsagent from sales of this magazine.
4. Assuming that the weekly demand remains the same, show that the mean weekly profit from sales of Statistics Weekly will be greater if the newsagent stocks only 4 copies.
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3. A game at a school fete is played with a fair coin and a random number generator which generates random integers between 1 and 52 inclusive. It costs 50 pence to play the game. First, the player tosses the coin. If it lands on tails, the player loses. If it lands on heads, the player is allowed to generate a random number. If the number is 1 , the player wins \(\pounds 5\). If the number is between 2 and 13 inclusive, the player wins \(\pounds 1\). If the number is greater than 13 , the player loses.

1. Find the probability distribution of the player's profit.
2. Find the mean and standard deviation of the player's profit.
3. Given that 200 people play the game, calculate
   1. the expected number of players who win some money,
   2. the expected profit for the fete.

5 Joanne has 10 identically-shaped discs, of which 1 is blue, 2 are green, 3 are yellow and 4 are red. She places the 10 discs in a bag and asks her friend David to play a game by selecting, at random and without replacement, two discs from the bag.

1. Show that:
   1. the probability that the two discs selected are the same colour is \(\frac { 2 } { 9 }\);
   2. the probability that exactly one of the two discs selected is blue is \(\frac { 1 } { 5 }\).
2. Using the discs, Joanne plays the game with David, under the following conditions: If the two discs selected by David are the same colour, she will pay him 135p. If exactly one of the two discs selected by David is blue, she will pay him 145p. Otherwise David will pay Joanne 45p.
   1. When a game is played, \(X\) is the amount, in pence, won by David. Construct the probability distribution for \(X\), in the form of a table.
   2. Show that \(\mathrm { E } ( X ) = 33\).
3. Joanne modifies the game so that the amount per game, \(Y\) pence, that she wins may be modelled by $$Y = 104 - 3 X$$
   1. Determine how much Joanne would expect to win if the game is played 100 times.
   2. Calculate the standard deviation of \(Y\), giving your answer to the nearest 1 p .