Quizzes and exercises of probability
These quizzes and exercises help us review and practice what we obtained in this chapter. They are presented in "show-hide" form similar to the sections of a web page as we are already familiar. Quizzes are multiple choice questions. After choosing an option, an announcement of result appears. To return back to this page, we click "OK" on the announcement. Of course we can choose again.
In exercises, we are required to calculate a value then fill it in an empty rectangle. Note that in this rectangle, only value is acceptable, its unit is not required. After filling the result, we click on "Answer". If the answer is correct, the border of the rectangle is green and there is green "V" symbol in the adjacent square. If the answer is wrong, the border of the rectangle is red and there is red "X" symbol in the adjacent square. We can erase the answer and retry by clicking on "Retry".
For each exercise, there is a hidden solution. To show this solution, we click on "Solution" tab. But try to solve exercises by ourself, don't abuse these solutions.
Quiz 1
Element x belongs to set A or set B, hence
Quiz 2
A and B are mutually exclusive events. Probability of the event (A∪B) is
Quiz 3
If probability of event A does not depend on the occurrence of B, A and B are
Quiz 4
Rolling 2 homogeneous dice, probability of the event "two upper faces are equal" is
Quiz 5
If two events are mutually exclusive, they will be
Quiz 6
A and B are two independent events and their probability are positive. Hence
Quiz 7
When `n>3`, `n!` is divisible by
Exercise 1
A food company would like to understand the preference of consumer about yoghurt. The first attribute is flavor with two options: pineapple and banana. The second one is net weight with two options: 80 g and 100 g. The survey consists of 200 customer and the result is summarized in Table 1. Numbers in this table are number of consumers corresponds to columns and rows.
| Pineapple | Banana | Total | |
|---|---|---|---|
| 80 g | 28 | 57 | 85 |
| 100 g | 47 | 68 | 115 |
| Total | 75 | 125 | 200 |
Take randomly a customer.
a. What is the probability that this customer prefers 100 g yoghurt ?
b. What is the probability that this customer prefers pineapple and 80 g yoghurt?
Solution
a. `P_"a"=115/200=0,575`
b. `P_"b"=28/200=0,14`
Exercise 2
The effectiveness of a healing method is 80%. There are two patients treated by this method. Assume that the results of these treatments are independent. What is the probability that this method is ineffective for both patients ?
Solution
Denote `P_"K"` is the probability that this healing method is ineffective: `P_"K"=0,2`
The probability which this method is ineffective for both patients is:
`P=P_"K"xxP_"K"= 0,2xx0,2=0,04`
Exercise 3
There are 7 books and we would like to arrange 3 books on the shelf in order. How many ways are possible ?
Solution
The number of possibilities is the permutation of 3 from 7:
`P(7,\ 3)=7xx(7-1)xx(7-2)=210`
Exercise 4
When A plays 10 games with B, A wins 5 games, loses 3 games, and draws 2 games. In a tournament, A and B play three games. Assume that the results of all the games are independent.
a. What is the probability that A wins all 3 games?
b. What is the probability that there are 2 games draw ?
c. What is the probability that these player win alternately?
Solution
In order to facilitate the reasoning, we use these symbol:
- A and B are the name of player (as above),
- each game is indexed by a number,
- draw game is symbolized as D, for example, "second game draw" is symbolized as D2
- "player P wins game `i`" is symbolized as P`i`, for example "B wins 1st game" is symbolized as B1
- "player P loses game `i`" is symbolized as Px`i`, for example "A loses 3rd game" is symbolized as Ax3
- "event does not occur" is indexed by "N", for example "A does not win 1st game" is symbolized as AN1, "B does not lose 2nd game" is symbolized as BNx2, "3rd game does not draw" is symbolize as DN3.
Hence in game `i` : `P("A"_i)=0,5` ; `P("A"_("N"i))=0,5` ; `P("A"_("x"i))=0,3` ; `P("A"_("Nx"i))=0,7` ; `P("B"_i)=0,3` ; `P("B"_("N"i))=0,7` ; `P("B"_("x"i))=0,5` ; `P("B"_("Nx"i))=0,5 ; P("D"_i)=0,2` ; `P("D"_("N"i))=0,8`.
a. The probability that A wins all 3 games is :
`P_"a"=P("A"_1\ "A"_2\ "A"_3)=0,5xx0,5xx0,5=0,125`
b. To obtain 2 games draw, there are 3 possibility: the 1st and 2nd game draw, the 3rd game does not draw, or the 1st and 3rd game draw, the 2nd game does not draw, or the 2nd and 3rd game draw, the 1st game does not draw.
The probability that there are two draw game is:
`P_"b"=P("D"_1\ "D"_2\ "D"_("N"3)) + P("D"_1\ "D"_("N"2)\ "D"_3)+P("D"_("N"1)\ "D"_2\ "D"_3)`
`P_"b"=(0,2xx0,2xx0,8)+(0,2xx0,8xx0,2)+(0,8xx0,2xx0,2)=0,096`
c. To obtain these players win alternately, there are 2 possibilities: A wins the 1st and 3rd games, B wins 2nd games or B wins 1st and 3rd games, A wins 2nd game.
Therefore the probability that these players win alternately is:
`P_"c"=P("A"_1\ "B"_2\ "A"_3)+P("B"_1\ "A"_2\ "B"_3)`
`P_"c"=(0,5xx0,3xx0,5)+(0,3xx0,5xx0,3)=0,12`
OK
This web page was last updated on 02 December 2018.