Question 1: There are movie tickets. Xiao Ming and Xiao Li drew lots to decide who could go to the movies, so they prepared two identical notes, one saying "Go" and the other saying "No". Whoever draws "Go" will go to the movies. Is this fair?
Question 2: We choose one of the three students to attend the concert by drawing lots. Prepare three identical sheets of paper in advance, mark them with 1, and leave the other two sheets blank. Is it fair to put three pieces of paper in a box and mix them evenly, and then let three students touch the paper?
Second, students discuss:
Classmate a and classmate b
Third, the question:
(1) Draw lots first, then draw lots. If the first smoker wins, the last smoker won't win. However, if the person who smokes first fails, the person who smokes later has a better chance?
(2) Is the probability of winning the lottery the same for the person who draws first and the person who draws later?
Fourth, teachers and students explore * * *:
Let's calculate the probability of each person winning the lottery:
Suppose these three students are marked as A, B and C, and the order of their drawing lots is: A first, B second and C third. Among the three small pieces of paper, the marked piece of paper is marked as A, and the remaining two unmarked pieces of paper are marked as and respectively.
We use a table to list all possible outcomes:
first time
(1) The second time
the third time
(c) All possible outcomes
begin
As can be seen from the above figure, there are six possible results when A, B and C draw lots in turn, and they are equally possible.
The two results of A and A are a lottery, and P (a lottery) = 1/3.
The result of a and a is b, P (B) = 1/3.
The result of A and A is the draw in C, and P (the draw in C) = 1/3.
Verb (abbreviation of verb) summary:
1, the lottery is a fair game.
Although the lottery comes first, the chances of winning the lottery are the same among those who draw first and those who draw later, so it is _ _ _ _ _ for everyone, and there is no need to earn the lottery back first.
Note: Random numbers generated by calculators or ball games are all mathematical models of lottery tickets, and these methods of selecting individuals are fair.
2. Drawing tree diagram or list method is an effective method to analyze fair game.
Sixth, example teaching
Example 1. Draw lots to choose two students from three classes to go to the movies. Is this method fair? Please provide a justification for the answer.
Example 2: Xiaoming and Xiao Li each throw a dice. If the sum of the points of two dice is odd, Xiao Ming gets one point, otherwise Xiao Li gets one point, and whoever gets ten points first wins. Is this game fair to both sides? The fairness of the game is that the probability of winning is equal. )
Example 3: Xiaolan and Tan Xiao determine the position of P(x, y) by throwing two dice A and B respectively. They stipulate that the number of points thrown by Xiaolan is X and the number of points thrown by Tan Xiao is Y. So, what is the probability that the number of points they determine falls on the known straight line y=-2x+6?
Example 4: Turn the two turntables once as shown in the figure.
(1) Find the probability that the pointer points to the red area once and then to the yellow area again.
(2) Please use these two turntables to design a fair game for both players.
Seven. abstract
The method of drawing lots is reasonable