Лекция: Example 1
A cubical die and a tetrahedral die are thrown together.
a) If X is the discrete random variable “total scored”, write down the probability distribution for X.
b) Find E(X ).
c) Find Var(X ).
A game is now played with the two dice. Anna has the cubical die and Beth has the tetrahedral die. They each gain points according to the following rules:
• If the number on both dice is greater than 3, then Beth gets 6 points.
• If the tetrahedral die shows 3 and the cubical die less than or equal to 3, then Beth gets 4 points.
• If the tetrahedral die shows 4 and the cubical die less than or equal to 3, then Beth gets 2 points.
• If the tetrahedral die shows a number less than 3 and the cubical die shows a 3, then Anna gets 5 points.
• If the tetrahedral die shows a number less than 3 and the cubical die shows a 1 or a 2, then Anna gets 3 points.
• If the tetrahedral die shows 3 and the cubical die greater than 3, then Anna gets 2 points.
• If the tetrahedral die shows a number less than 3 and the cubical die shows a number greater than 3, then Anna gets 1 point.
d) Write out the probability distribution for Y, “the number of points gained by Anna”.
e) Calculate E(Y ) and Var(Y ).
f) The game is now to be made fair by changing the number of points Anna gets when the tetrahedral die shows a number less than 3 and the cubical die shows a 1 or a 2. What is this number of points to the nearest whole number?
a) A probability space diagram is the easiest way to show the possible outcomes.
Hence the probability distribution for X is:
| x |
| P(X=x) |
b) Since the probability distribution is symmetrical E(X)=6.
c)
d)By considering the possibility space diagram again the probability distribution is:
| y | -6 | -4 | -2 |
| P(Y=y) |
f)Let the number of points Anna gains when the tetrahedral die shows a number less than 3 and the cubical die shows a 1 or a 2 be y.
In this case the probability distribution is now:
| y | -6 | -4 | -2 | y |
| P(Y=y) |
Since the game is now fair,