A game consists of throwing darts onto a target. The target is divided into four sectors as shown in the figure below

Assume that the throws are independent from one another and that the players hit the target on every throw.

1. A player throws a dart. Let represent the probability of getting point, the probability of getting points, and the probability of getting points.

Thus, . Given and , find the values of and .

2. One of the games consists of throwing a maximum of three darts. The player wins the game if she obtains a total (for the three throws) of or more points. If she has a total of or more points after two throws, she does not throw the third dart.

Let represent the event “the player wins the game in throws”, the event “the player wins the game in throws”, and the event “the player loses the game”. Let represent the probability of an event .

(a) Show that by using a possibility tree

(b) Derive knowing that

3. A player plays six games according to the rules given in Question 2. What is the probability that she will win at least one game?

4. The price for one game is fixed at euros. If the player wins the game in two throws, she receives euros. If she wins in three throws, she receives euros. If she loses, she receives nothing. Let be the random variable that represents the algebraic gain by the player for one game. The possible values of are therefore: and .

(a) Give the probability distribution of

(b) Determine the expected value of . Is the game favorable to the player?

Source: Baccalaureat General, Session Avril 2011, Pondichery, Serie Scientifique, www.ilemaths.net

SOLUTION

1. Find the values of and .

We have three unknown variables and three equations

Compare Eq. and

Multiply both sides by

Substitute the values of and into Eq.

Substitute the value of into Eq.

Substitute the value of into Eq.

2. (a) Show that by using a possibility tree

We will draw the tree symbolically by writing the scores of a event as follows

, first throw 5 points; second throw 5 points

, first throw 5 points; second throw 3 points

, first throw 3 points; second throw 5 points

2. (b) Derive knowing that

3. A player plays six games according to the rules given in question 2.

What is the probability that she will win at least one game?

Winning at least one game means winning or games. In terms of losing that means losing or games.

Probability of losing all six games

Probability of winning at least one game

4. (a) Give the probability distribution of

when the player loses the game

when the player wins the game in three throws

when the player wins the game in two throws

4. (b) Determine the expected value of . Is the game favorable to the player?

The negative expected value indicates that the player is expected to lose money over the long run. The game is not favorable to the player.

**Answer**

1.

2. (a) see proof in solution

2. (b)

3. Probability of winning at least one game

4. (a)

4. (b) ; game not favorable to player