BACCALAURÉAT GÉNÉRAL ET TECHNOLOGIQUE SESSION 2009 ÉPREUVE SPÉCIFIQUE MENTION « SECTION EUROPÉENNE OU DE LANGUE ORIENTALE »
Académies de Paris-Créteil-Versailles Binôme : Anglais / Mathématiques Sujet n° 12
Probability The first part of this page is a summary that can be helpful to do the exercise.
When finding probabilities a tree diagram can be used to list the sample space. Example: a biased coin is tossed twice, a tree diagram of this experiment is: 0.6 heads If X is a variable with values x1 , x 2 , … x n tails heads 0.6 and P the probability associated to this variable, 0.4
then the expectation (or expected value) of X is:
E ( X ) = x1 p1 + x 2 p 2 + K + x n p n . 0.6 0.4
tails
heads
In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.
tails
0.4
P(obtaining one heads and one tails) = 0.6 × 0.4 + 0.4 × 0.6 P(not obtaining two heads) = 1 - 0.6 × 0.6
Exercise At a fair, a game is as follows: The player puts £1 on one of 6 cases numbered from 1 to 6 (his stake is £1); The stallholder rolls two dice whose sides are numbered from 1 to 6; If the number chosen by the player appears on one dice only, then the player wins £2 (and takes his stake back); if it appears on both dice then he wins £3; if it does not appear at all, he loses his stake. We suppose that the dice is not biased, that is each side is equally likely to be rolled. • • •
1. Find the probabilities of the following events : A: “the player loses his stake”; B: “ he wins £2”; C: “ he wins £3”. If you choose to draw a tree diagram, then it should have as few branches as possible. 2. Find the player’s expectation of gain. 3. A player tries his luck twice. a) Draw a tree diagram with the possible gains and losses. b) Find the probabilities of the following events: D: “he loses £2”; E: “he wins”; F: “he wins £5”. 4. A player tries his luck three times. Add carefully a few branches to the tree diagram drawn for question 3 and find the probability that the player neither wins nor loses any money. Give the answer as a percentage. Page : 1/1