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SIXTH GRADE / PROBABILITY / TEST 5.1.2

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH.
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

SOME OF OUR QUESTIONS ARE REALLY HARD, FOR THOSE THAT ARE SKILLED IN MATH.
SKIP THEM IN CASE NOT FOR YOU.

IF YOU DESIRE SO, WE CAN DISCUSS HARD/INTERESTING QUESTIONS IN OUR FORUM.

 PROBABILITY
    ♦ 
Sometimes an event depends on very many parameters and some of them are unknown. (Theory of) Probability is our partly sufficient but necessary response to such situations.
    ♦ If we throw a coin in the air we can not control the forces on the coin and do not know if the outcome will be head or tail. But we know if we throw 1000 times 500 of each is to be expected.
    ♦ Experiment, outcome, sample space, event, independent event, discrete/ continuous space are important concepts to be known.
    Ref.-1
    Ref.-2
1. A dice is rolled and a coin is tossed both twice. What is the probability that the dice has a total of 12 and the coin shows 2 heads ?

a) 1/36
b) 1/144
c) 1/72
d) 1/288
All are independent trials, probabilities are to be multiplied: 1/6·1/6·1/2·1/2 = ...
2. A dice is rolled and a coin is tossed and they got collided before final result. What is the probability that they show a "4" and an "head" ?

a) 3/21
b) 1/6
c) 1/8
d) 1/12
They are still independent events according to our theory. The effects of collision are to be added to other very many unknown factors .
3. What is the probability of getting "K of Hearts" from (the right) half a deck of cards (26) shown in the figure ?

!


a) 1/52
b) 1/26
c) 1/48
d) 2/47
Half or a quarter does not change anything, there are a total of 52 cards without any xtra info .
4. What is the probability of getting "Q of Hearts" from the right half of a deck of cards (26) shown in the figure ?

!


a) 1/104
b) 1/52
c) 1/51
d) 1/26
"K of Hearts" from the other half is opened .
5. What is the probability of not finding "Q and J of Hearts" in the right half of a deck of cards (26) shown in the figure given in the previous question?

a) 1/2
b) 25/102
c) 1/4
d) 24/51
Not finding the first has a probability of 25/51 . If it is found, the probability of not finding the second is 25/50.
y
6. Assume that there are n (other) people in the room you are, and you are born on Tuesday. What is the mathematical expression for the probability that no one else in the room shares Tuesday as his/her birthday (of the week) with you ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) (6/7)n-1
b) (6/7)n
c) (6/7)n+1
d) (6/7)2·n


(6/7).. . Have please a look at ry1 .
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7. On a game show an entrant is given the choice of three doors. Behind one door is a car and behind the other two doors are muppets. The entrant picks a door and the host, who knows what behind each door exists, opens one of the other "from his point equally likely" doors to show a muppet. Then the host asks the entrant if he wants to change his choice.
Should the entrant switch ?

!


a) Probably
b) Of course, ..
c) Noo , ...
d) Should try
If "from his point equally likely" and he knows the contents ... .
8. A box has three drawers; one contains two gold coins, one contains two silver coins, and one contains one gold coin and one silver coin. Assume that one drawer is selected randomly and that a randomly selected coin from that drawer turns out to be gold. What is the probability that the chosen drawer is one that contains two gold coins ?

!


a) 1/3
b) 1/4
c) 3/4
d) 2/3
There are 3 gold coins and finding one of them has the probability of 1/3 .
The formal solution to this problem are in : http://en.wikipedia.org/wiki/Bertrand's_box_paradox and
http://www.davidson.edu/academic/economics/martin/InterestingProbabilityQuestions.pdf .
9. Assume that there are 20 people (including yourself) in the room. Ignoring leap years, what is the probability that at least one "other" person in the room shares your birthday ?

a) 1-(364/365)19
b) 1-(364/365)20
c) 1-(365/366)19
d) 1-(365/366)20
The case of no other person shares is similar to problem 6 : (364/365)19 .
10. Assume that there are 20 people (including yourself) in the room and one of them has birthday on Feb. 29 . What is the probability that at least one other person in the room shares his birthday ?

a) (1460/1461)20
b) (1462/1461)19
c) (1460/1461)19
d) (1460/1462)19
4x365+1 = 1461 and 20-1 = 19 ... .
11. The dodecahedron in the following figure is rolled. What is the probability that the upper face is a number which is divisible by 2 or by 5 ?

!


a) 5/12
b) 7/12
c) 1/3
d) 1/4
divisible by 2 : 2,4,6,8,10,12
divisible by 5 : 5,10 .
y
12. In a given room there are N<13 people. What is the mathematical formula for the probability that no two people share the same birth-month ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2".

RY1 RY2



a) 11·10·...·[11-(N-1)]/12N-1
b) 11·10·...·[12-(N-1)]/12N-1
c) 11·10·...·[12-(N-1)]/12N
d) 12·11·...·[12-(N-1)]/12N-1
If N=1, certain event and if N=13, impossible event .
The first has a definite month of birth; the probability that the second does not share is 11/12. For further help have please a look at ry1 .
13. In a classroom there are 12 boys and 9 girls. Three of them get out of the room. What is the rounded probability that all are boys ?

!


a) 18%
b) 21%
c) 14%
d) 17%
The probability that first is a boy is 12/21. The second has a probability of 11/20 and the third ... .
12/21 · 11/20 · 10/19 = 0,165(41353383458646616541353383459) .
14. Which of the 6 sections seen in the figure below is most likely to be completed ?

!


a) lower left
b) middle left
c) upper left
d) upper right


The one that has most at present .
15. What is the probability that the next 2 draws will be numbers on upper right of the figure of the previous question ?

a) 2/15·7/59
b) 2/13·7/59
c) 2/15·7/57
d) 1/15·7/57
There are 60 numbers not drawn .
The probability of the first hit is 2/15 .
16. Player 1, 2 and 3 are to open the 3 reversed cards. What is the probability that the first two do not open the card numbered "1" ?

!


a) 1/3
b) 1/2
c) 2/3
d) 3/5
Then the 3rd opens it .
17. The rest of the deck, shown at left in the figure given in question-16, has 47 cards. What is the probability of finding King of Diamaonds in 2 draws ?

a) 77/1571
b) 93/2162
c) 1/47 ·1/46
d) 1/45 ·1/46
1/47 + 1/46 = ?
y
18. Assume that there are n<13 people in the room and P is the probability that at least two people in the room share the same month of birth. In which of the following intervals is the number n that makes P equal to 1/2 ?
In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2


a) 5 > n > 6
b) 4 > n > 5
c) 5 < n < 6
d) 4 < n < 5
n = ? if P = 1 - (11/12)·(10/12)·...·[(12-n)/12] = 1/2. Have please a look at ry1 .
19. A particle enters into the grid shown at time t=0 . All matches are of unit length and the particle can only diffuse from the other end to the head with unit velocity, i.e. it travels through the match in a second (only in the given direction).
What will be the probability of finding the particle just having passed point B and traveling horizontally just a bit after t = 4 s. if in case of permitted directions, traveling through probabilities are equal at each node ?

!


a) 1/8
b) 1/4
c) 1/16
d) 1/12
At each node there are 2 possible ways each with 1/2 probability .
20. The same cast as in the previous question but the particle is at node B at t=0 . What is the probability that it will be at the node of (only) two heads at t=4 (s.) ?

a) 1
b) 1/2
c) 1/16
d) 1/32
Initially it can travel horizontally or vertically with equal probabilities .
All ways go to Rome, it sholuld move and no other way to go .
21. A dice is rolled thrice. What will be the probability that all faces will appear to be odd ?

a) 1/4
b) 1/12
c) 1/8
d) 2/33
sample space:
{{1,1,1},{1,1,2},{1,1,3},{1,1,4},{1,1,5},{1,1,6},
{1,2,1},{1,2,2},{1,2,3},{1,2,4},{1,2,5},{1,2,6},
{1,3,1},{1,3,2},{1,3,3},{1,3,4},{1,3,5},{1,3,6},
{1,4,1},{1,4,2},{1,4,3},{1,4,4},{1,4,5},{1,4,6},
{1,5,1},{1,5,2},{1,5,3},{1,5,4},{1,5,5},{1,5,6},
{1,6,1},{1,6,2},{1,6,3},{1,6,4},{1,265},{1,6,6},..., and these all repeat 6 times with the same structure giving altogether 6x36 = 216 elements up to {6,6,6}}
We of course need to be careful about {2,.,.},{4,.,.} and {6,.,.} series ... This means we are to get only 3·9 = ... times all odd in the sample space.
1/2·1/2·1/2 = ?
y
22. E1 and E2 are 2 events in a definite events set which consists of all subsets of a sample space with P(E1) = 2/3, P(E2) = 5/12 and P(E1∪E2) = 8/12 . Which of the followings is then false ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .

RY1 RY2



a) P(˜E1) = 1/3
b) P[˜(E1∩E2)] = 1/2
c) P(˜E2) = 7/12
d) P[(E1∪E2)] = 2/3


˜ means negation, P(˜E1) is the probability of not happening of E1 . For further help, have please a look at ry1.
23. Nathan and Johahn are to select a person from the following picture independently. What is the probability that they choose the same person from the back row ?

!


a) 1/9
b) (2/9)3
c) (5/9)2
d) (5/9)·(1/9)
There are 9 people and 5 of them are at the back row. First chooses one from the ack row with P=5/9.
24. 7 of the people in the picture given in the previous question have driving licenses. What is the probability that a random choice will spot a man with a driving license ?

a) (5/9)/(7/9)2
b) (5/9)·(7/9)2
c) (5/9)/(7/9)
d) (5/9)·(7/9)
The probability of spotting a man is 5/9 and ...
25. The following picture indicates the number of bacteria at different regions of a plate. What is the probability that a randomly chosen one will be from the center, red colored region, of the plate ?

!


a) 2/503
b) 1/294
c) 1/273
d) 1/243
Here it is important that you evaluate the sum via 1+4 = 5; 2x5=10; 10+7=17; 2x17=34 and 16+4=20; 2x20=40; 40+26=66; 2x66=132 and ... .
This is the feedback!

THANK YOU FOR YOUR VISIT; HOPE WE COULD BE OF USE.
FOR MUTUAL BENEFIT YOU MAY USE OUR FORUM.
© 2012 , İsmail GERMAN. ALL RIGHTS RESERVED
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